Polynomials - Ninth Grade Math
1. Polynomial Vocabulary
Polynomial: An algebraic expression with one or more terms containing variables with non-negative integer exponents
Etymology: "Poly" (many) + "nomial" (terms)
General Form:
$$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$$
where $a_n, a_{n-1}, \ldots, a_0$ are coefficients and $n$ is a non-negative integer
Etymology: "Poly" (many) + "nomial" (terms)
General Form:
$$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$$
where $a_n, a_{n-1}, \ldots, a_0$ are coefficients and $n$ is a non-negative integer
Term: A single part of a polynomial separated by + or - signs
• Example: In $3x^2 + 5x - 7$, there are three terms: $3x^2$, $5x$, and $-7$
Coefficient: The numerical factor of a term
• In $5x^2$, the coefficient is 5
Constant Term: A term without a variable
• In $2x^2 + 3x - 4$, the constant is $-4$
Leading Term: The term with the highest degree
Leading Coefficient: The coefficient of the leading term
• Example: In $3x^2 + 5x - 7$, there are three terms: $3x^2$, $5x$, and $-7$
Coefficient: The numerical factor of a term
• In $5x^2$, the coefficient is 5
Constant Term: A term without a variable
• In $2x^2 + 3x - 4$, the constant is $-4$
Leading Term: The term with the highest degree
Leading Coefficient: The coefficient of the leading term
Degree of a Term: The sum of exponents of all variables in that term
• $5x^3$ has degree 3
• $4x^2y$ has degree 3 (2+1)
• $7$ has degree 0 (constant)
Degree of a Polynomial: The highest degree among all its terms
• $3x^4 + 2x^2 - 5x + 1$ has degree 4
• $5x^3$ has degree 3
• $4x^2y$ has degree 3 (2+1)
• $7$ has degree 0 (constant)
Degree of a Polynomial: The highest degree among all its terms
• $3x^4 + 2x^2 - 5x + 1$ has degree 4
Types of Polynomials by Number of Terms
| Name | Number of Terms | Example |
|---|---|---|
| Monomial | One term | $5x^3$, $-7$, $2xy$ |
| Binomial | Two terms | $3x + 5$, $x^2 - 4$ |
| Trinomial | Three terms | $x^2 + 5x + 6$, $2a^2 - 3a + 1$ |
| Polynomial | More than three terms | $x^3 + 2x^2 - 5x + 7$ |
Types of Polynomials by Degree
| Degree | Name | General Form | Example |
|---|---|---|---|
| 0 | Constant | $a$ | $5$, $-3$ |
| 1 | Linear | $ax + b$ | $2x + 3$ |
| 2 | Quadratic | $ax^2 + bx + c$ | $x^2 - 4x + 4$ |
| 3 | Cubic | $ax^3 + bx^2 + cx + d$ | $2x^3 + x^2 - 5$ |
| 4 | Quartic | $ax^4 + bx^3 + cx^2 + dx + e$ | $x^4 - 16$ |
| 5 | Quintic | $ax^5 + ...$ | $x^5 + 2x^3 - 1$ |
What is NOT a Polynomial:
• Negative exponents: $x^{-2} + 3$
• Fractional exponents: $x^{1/2} + 5$
• Variables in denominator: $\frac{1}{x} + 2$
• Variables under radical: $\sqrt{x} + 1$
• Negative exponents: $x^{-2} + 3$
• Fractional exponents: $x^{1/2} + 5$
• Variables in denominator: $\frac{1}{x} + 2$
• Variables under radical: $\sqrt{x} + 1$
2-3. Model Polynomials with Algebra Tiles
Algebra Tiles: Visual manipulatives used to represent algebraic expressions
Tile Representations:
• Large square: $x^2$ (x by x)
• Rectangle: $x$ (x by 1)
• Small square: $1$ (1 by 1)
• Negative tiles: Shown in different color or shaded
• Zero Pairs: One positive and one negative tile cancel out
Tile Representations:
• Large square: $x^2$ (x by x)
• Rectangle: $x$ (x by 1)
• Small square: $1$ (1 by 1)
• Negative tiles: Shown in different color or shaded
• Zero Pairs: One positive and one negative tile cancel out
Using Algebra Tiles:
For $x^2 + 3x + 2$:
• 1 large square ($x^2$)
• 3 rectangles ($3x$)
• 2 small squares ($2$)
For $2x^2 - x + 3$:
• 2 large squares positive
• 1 rectangle negative
• 3 small squares positive
For $x^2 + 3x + 2$:
• 1 large square ($x^2$)
• 3 rectangles ($3x$)
• 2 small squares ($2$)
For $2x^2 - x + 3$:
• 2 large squares positive
• 1 rectangle negative
• 3 small squares positive
Adding/Subtracting with Tiles:
• Combine like tiles
• Use zero pairs to eliminate opposites
• Count remaining tiles to get answer
• Combine like tiles
• Use zero pairs to eliminate opposites
• Count remaining tiles to get answer
4-5. Add and Subtract Polynomials
Like Terms: Terms with the same variable(s) raised to the same power(s)
• $3x^2$ and $-5x^2$ are like terms
• $2x$ and $7x$ are like terms
• $4x^2$ and $4x$ are NOT like terms
• $3x^2$ and $-5x^2$ are like terms
• $2x$ and $7x$ are like terms
• $4x^2$ and $4x$ are NOT like terms
Rule for Adding/Subtracting Polynomials:
• Combine coefficients of like terms only
• Variables and exponents stay the same
• Cannot combine unlike terms
$$ax^n + bx^n = (a + b)x^n$$
$$ax^n - bx^n = (a - b)x^n$$
• Combine coefficients of like terms only
• Variables and exponents stay the same
• Cannot combine unlike terms
$$ax^n + bx^n = (a + b)x^n$$
$$ax^n - bx^n = (a - b)x^n$$
Addition of Polynomials
Steps to Add Polynomials (Horizontal Method):
Step 1: Remove parentheses
Step 2: Identify and group like terms
Step 3: Add coefficients of like terms
Step 4: Write answer in standard form (highest to lowest degree)
Step 1: Remove parentheses
Step 2: Identify and group like terms
Step 3: Add coefficients of like terms
Step 4: Write answer in standard form (highest to lowest degree)
Example 1: Add $(3x^2 + 5x - 2) + (2x^2 - 3x + 7)$
Step 1: Remove parentheses:
$3x^2 + 5x - 2 + 2x^2 - 3x + 7$
Step 2: Group like terms:
$(3x^2 + 2x^2) + (5x - 3x) + (-2 + 7)$
Step 3: Combine:
$5x^2 + 2x + 5$
Answer: $5x^2 + 2x + 5$
Step 1: Remove parentheses:
$3x^2 + 5x - 2 + 2x^2 - 3x + 7$
Step 2: Group like terms:
$(3x^2 + 2x^2) + (5x - 3x) + (-2 + 7)$
Step 3: Combine:
$5x^2 + 2x + 5$
Answer: $5x^2 + 2x + 5$
Example 2: Add $(4x^3 - 2x + 1) + (x^3 + 5x^2 + 3x - 4)$
Group: $(4x^3 + x^3) + (5x^2) + (-2x + 3x) + (1 - 4)$
Combine: $5x^3 + 5x^2 + x - 3$
Answer: $5x^3 + 5x^2 + x - 3$
Group: $(4x^3 + x^3) + (5x^2) + (-2x + 3x) + (1 - 4)$
Combine: $5x^3 + 5x^2 + x - 3$
Answer: $5x^3 + 5x^2 + x - 3$
Subtraction of Polynomials
Steps to Subtract Polynomials:
Step 1: Distribute the negative sign to all terms in second polynomial
Step 2: Change subtraction to addition
Step 3: Group like terms
Step 4: Combine like terms
Step 5: Write in standard form
Step 1: Distribute the negative sign to all terms in second polynomial
Step 2: Change subtraction to addition
Step 3: Group like terms
Step 4: Combine like terms
Step 5: Write in standard form
Example 3: Subtract $(5x^2 + 3x - 4) - (2x^2 + x - 6)$
Step 1: Distribute negative:
$5x^2 + 3x - 4 - 2x^2 - x + 6$
Step 2: Group like terms:
$(5x^2 - 2x^2) + (3x - x) + (-4 + 6)$
Step 3: Combine:
$3x^2 + 2x + 2$
Answer: $3x^2 + 2x + 2$
Step 1: Distribute negative:
$5x^2 + 3x - 4 - 2x^2 - x + 6$
Step 2: Group like terms:
$(5x^2 - 2x^2) + (3x - x) + (-4 + 6)$
Step 3: Combine:
$3x^2 + 2x + 2$
Answer: $3x^2 + 2x + 2$
Example 4: Subtract $(7x^3 - 4x + 2) - (3x^3 + 2x^2 - 5x + 1)$
Distribute: $7x^3 - 4x + 2 - 3x^3 - 2x^2 + 5x - 1$
Group: $(7x^3 - 3x^3) + (-2x^2) + (-4x + 5x) + (2 - 1)$
Combine: $4x^3 - 2x^2 + x + 1$
Answer: $4x^3 - 2x^2 + x + 1$
Distribute: $7x^3 - 4x + 2 - 3x^3 - 2x^2 + 5x - 1$
Group: $(7x^3 - 3x^3) + (-2x^2) + (-4x + 5x) + (2 - 1)$
Combine: $4x^3 - 2x^2 + x + 1$
Answer: $4x^3 - 2x^2 + x + 1$
Vertical Method
Example 5: Add vertically: $(3x^2 + 2x - 5) + (x^2 - 4x + 3)$
Answer: $4x^2 - 2x - 2$
3x² + 2x - 5
+ x² - 4x + 3
_______________
4x² - 2x - 2
Answer: $4x^2 - 2x - 2$
Application: Finding Perimeter
Example 6: A rectangle has length $(3x + 5)$ and width $(x + 2)$. Find perimeter.
Formula: $P = 2l + 2w$
$P = 2(3x + 5) + 2(x + 2)$
$P = 6x + 10 + 2x + 4$
$P = 8x + 14$
Answer: Perimeter = $8x + 14$
Formula: $P = 2l + 2w$
$P = 2(3x + 5) + 2(x + 2)$
$P = 6x + 10 + 2x + 4$
$P = 8x + 14$
Answer: Perimeter = $8x + 14$
6. Multiply a Polynomial by a Monomial
Distributive Property:
$$a(b + c) = ab + ac$$
For Polynomials:
$$a(b + c + d) = ab + ac + ad$$
Multiply each term of the polynomial by the monomial
$$a(b + c) = ab + ac$$
For Polynomials:
$$a(b + c + d) = ab + ac + ad$$
Multiply each term of the polynomial by the monomial
Steps:
Step 1: Distribute the monomial to each term inside parentheses
Step 2: Multiply coefficients
Step 3: Add exponents when multiplying same bases
Step 4: Simplify
Step 1: Distribute the monomial to each term inside parentheses
Step 2: Multiply coefficients
Step 3: Add exponents when multiplying same bases
Step 4: Simplify
Example 1: $3x(2x^2 + 5x - 4)$
Distribute:
$= 3x \cdot 2x^2 + 3x \cdot 5x + 3x \cdot (-4)$
$= 6x^3 + 15x^2 - 12x$
Answer: $6x^3 + 15x^2 - 12x$
Distribute:
$= 3x \cdot 2x^2 + 3x \cdot 5x + 3x \cdot (-4)$
$= 6x^3 + 15x^2 - 12x$
Answer: $6x^3 + 15x^2 - 12x$
Example 2: $-4x^2(3x^2 - 2x + 5)$
$= -4x^2 \cdot 3x^2 + (-4x^2) \cdot (-2x) + (-4x^2) \cdot 5$
$= -12x^4 + 8x^3 - 20x^2$
Answer: $-12x^4 + 8x^3 - 20x^2$
$= -4x^2 \cdot 3x^2 + (-4x^2) \cdot (-2x) + (-4x^2) \cdot 5$
$= -12x^4 + 8x^3 - 20x^2$
Answer: $-12x^4 + 8x^3 - 20x^2$
Example 3: $5xy(2x^2 + 3xy - y^2)$
$= 10x^3y + 15x^2y^2 - 5xy^3$
Answer: $10x^3y + 15x^2y^2 - 5xy^3$
$= 10x^3y + 15x^2y^2 - 5xy^3$
Answer: $10x^3y + 15x^2y^2 - 5xy^3$
7-9. Multiply Two Binomials
FOIL Method: A technique for multiplying two binomials
F - First terms
O - Outer terms
I - Inner terms
L - Last terms
F - First terms
O - Outer terms
I - Inner terms
L - Last terms
FOIL Formula:
$$(a + b)(c + d) = ac + ad + bc + bd$$
where:
• $ac$ = First terms
• $ad$ = Outer terms
• $bc$ = Inner terms
• $bd$ = Last terms
$$(a + b)(c + d) = ac + ad + bc + bd$$
where:
• $ac$ = First terms
• $ad$ = Outer terms
• $bc$ = Inner terms
• $bd$ = Last terms
Steps Using FOIL:
Step 1: Multiply FIRST terms of each binomial
Step 2: Multiply OUTER terms
Step 3: Multiply INNER terms
Step 4: Multiply LAST terms
Step 5: Combine like terms
Step 1: Multiply FIRST terms of each binomial
Step 2: Multiply OUTER terms
Step 3: Multiply INNER terms
Step 4: Multiply LAST terms
Step 5: Combine like terms
Example 1: $(x + 3)(x + 5)$
F: $x \cdot x = x^2$
O: $x \cdot 5 = 5x$
I: $3 \cdot x = 3x$
L: $3 \cdot 5 = 15$
Combine: $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$
Answer: $x^2 + 8x + 15$
F: $x \cdot x = x^2$
O: $x \cdot 5 = 5x$
I: $3 \cdot x = 3x$
L: $3 \cdot 5 = 15$
Combine: $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$
Answer: $x^2 + 8x + 15$
Example 2: $(2x - 3)(x + 4)$
F: $2x \cdot x = 2x^2$
O: $2x \cdot 4 = 8x$
I: $-3 \cdot x = -3x$
L: $-3 \cdot 4 = -12$
Combine: $2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12$
Answer: $2x^2 + 5x - 12$
F: $2x \cdot x = 2x^2$
O: $2x \cdot 4 = 8x$
I: $-3 \cdot x = -3x$
L: $-3 \cdot 4 = -12$
Combine: $2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12$
Answer: $2x^2 + 5x - 12$
Example 3: $(3x - 2)(2x - 5)$
F: $6x^2$
O: $-15x$
I: $-4x$
L: $+10$
Answer: $6x^2 - 19x + 10$
F: $6x^2$
O: $-15x$
I: $-4x$
L: $+10$
Answer: $6x^2 - 19x + 10$
Special Cases: Perfect Square Patterns
Square of a Sum:
$$(a + b)^2 = a^2 + 2ab + b^2$$
Square of a Difference:
$$(a - b)^2 = a^2 - 2ab + b^2$$
Pattern: Square first, twice the product, square last
$$(a + b)^2 = a^2 + 2ab + b^2$$
Square of a Difference:
$$(a - b)^2 = a^2 - 2ab + b^2$$
Pattern: Square first, twice the product, square last
Example 4: $(x + 4)^2$
Method 1 - Pattern:
$(x + 4)^2 = x^2 + 2(x)(4) + 4^2$
$= x^2 + 8x + 16$
Method 2 - FOIL:
$(x + 4)(x + 4)$
$= x^2 + 4x + 4x + 16$
$= x^2 + 8x + 16$
Answer: $x^2 + 8x + 16$
Method 1 - Pattern:
$(x + 4)^2 = x^2 + 2(x)(4) + 4^2$
$= x^2 + 8x + 16$
Method 2 - FOIL:
$(x + 4)(x + 4)$
$= x^2 + 4x + 4x + 16$
$= x^2 + 8x + 16$
Answer: $x^2 + 8x + 16$
Example 5: $(3x - 5)^2$
$(3x)^2 - 2(3x)(5) + 5^2$
$= 9x^2 - 30x + 25$
Answer: $9x^2 - 30x + 25$
$(3x)^2 - 2(3x)(5) + 5^2$
$= 9x^2 - 30x + 25$
Answer: $9x^2 - 30x + 25$
Special Cases: Difference of Squares
Difference of Squares Pattern:
$$(a + b)(a - b) = a^2 - b^2$$
Key Feature: Middle terms cancel out!
$$(a + b)(a - b) = a^2 - b^2$$
Key Feature: Middle terms cancel out!
Example 6: $(x + 7)(x - 7)$
Using Pattern:
$= x^2 - 7^2$
$= x^2 - 49$
Verify with FOIL:
$= x^2 - 7x + 7x - 49$
$= x^2 - 49$ ✓
Answer: $x^2 - 49$
Using Pattern:
$= x^2 - 7^2$
$= x^2 - 49$
Verify with FOIL:
$= x^2 - 7x + 7x - 49$
$= x^2 - 49$ ✓
Answer: $x^2 - 49$
Example 7: $(2x + 3)(2x - 3)$
$= (2x)^2 - 3^2$
$= 4x^2 - 9$
Answer: $4x^2 - 9$
$= (2x)^2 - 3^2$
$= 4x^2 - 9$
Answer: $4x^2 - 9$
10-12. Multiply Polynomials
General Rule: Multiply each term of the first polynomial by each term of the second polynomial
Distributive Property (Extended):
$$(a + b)(c + d + e) = a(c + d + e) + b(c + d + e)$$
$$= ac + ad + ae + bc + bd + be$$
$$(a + b)(c + d + e) = a(c + d + e) + b(c + d + e)$$
$$= ac + ad + ae + bc + bd + be$$
Box Method (Area Model)
Example 1: $(x + 3)(2x^2 + 5x - 4)$ using box method
Create grid:
Add all products:
$2x^3 + 5x^2 + 6x^2 - 4x + 15x - 12$
$= 2x^3 + 11x^2 + 11x - 12$
Answer: $2x^3 + 11x^2 + 11x - 12$
Create grid:
| $2x^2$ | $+5x$ | $-4$ | |
|---|---|---|---|
| $x$ | $2x^3$ | $5x^2$ | $-4x$ |
| $+3$ | $6x^2$ | $15x$ | $-12$ |
Add all products:
$2x^3 + 5x^2 + 6x^2 - 4x + 15x - 12$
$= 2x^3 + 11x^2 + 11x - 12$
Answer: $2x^3 + 11x^2 + 11x - 12$
Distributive Method
Steps:
Step 1: Distribute first term of first polynomial to all terms of second
Step 2: Distribute second term of first polynomial to all terms of second
Step 3: Continue for all terms
Step 4: Combine like terms
Step 1: Distribute first term of first polynomial to all terms of second
Step 2: Distribute second term of first polynomial to all terms of second
Step 3: Continue for all terms
Step 4: Combine like terms
Example 2: $(x + 2)(x^2 - 3x + 4)$
Distribute $x$:
$x(x^2 - 3x + 4) = x^3 - 3x^2 + 4x$
Distribute $2$:
$2(x^2 - 3x + 4) = 2x^2 - 6x + 8$
Combine:
$x^3 - 3x^2 + 4x + 2x^2 - 6x + 8$
$= x^3 - x^2 - 2x + 8$
Answer: $x^3 - x^2 - 2x + 8$
Distribute $x$:
$x(x^2 - 3x + 4) = x^3 - 3x^2 + 4x$
Distribute $2$:
$2(x^2 - 3x + 4) = 2x^2 - 6x + 8$
Combine:
$x^3 - 3x^2 + 4x + 2x^2 - 6x + 8$
$= x^3 - x^2 - 2x + 8$
Answer: $x^3 - x^2 - 2x + 8$
Example 3: $(2x - 1)(x^2 + 3x - 5)$
$= 2x(x^2 + 3x - 5) - 1(x^2 + 3x - 5)$
$= 2x^3 + 6x^2 - 10x - x^2 - 3x + 5$
$= 2x^3 + 5x^2 - 13x + 5$
Answer: $2x^3 + 5x^2 - 13x + 5$
$= 2x(x^2 + 3x - 5) - 1(x^2 + 3x - 5)$
$= 2x^3 + 6x^2 - 10x - x^2 - 3x + 5$
$= 2x^3 + 5x^2 - 13x + 5$
Answer: $2x^3 + 5x^2 - 13x + 5$
Application: Finding Area
Example 4: A rectangle has length $(x + 5)$ and width $(x + 3)$. Find area.
Formula: $A = l \times w$
$A = (x + 5)(x + 3)$
$A = x^2 + 3x + 5x + 15$
$A = x^2 + 8x + 15$
Answer: Area = $x^2 + 8x + 15$ square units
Formula: $A = l \times w$
$A = (x + 5)(x + 3)$
$A = x^2 + 3x + 5x + 15$
$A = x^2 + 8x + 15$
Answer: Area = $x^2 + 8x + 15$ square units
13. Divide Polynomials by Monomials
Rule: Divide each term of the polynomial by the monomial
$$\frac{a + b + c}{d} = \frac{a}{d} + \frac{b}{d} + \frac{c}{d}$$
Exponent Rule:
$$\frac{x^m}{x^n} = x^{m-n}$$
$$\frac{a + b + c}{d} = \frac{a}{d} + \frac{b}{d} + \frac{c}{d}$$
Exponent Rule:
$$\frac{x^m}{x^n} = x^{m-n}$$
Steps:
Step 1: Write as separate fractions for each term
Step 2: Divide coefficients
Step 3: Subtract exponents for same variables
Step 4: Simplify each term
Step 1: Write as separate fractions for each term
Step 2: Divide coefficients
Step 3: Subtract exponents for same variables
Step 4: Simplify each term
Example 1: $\frac{12x^3 + 8x^2 - 4x}{4x}$
Separate:
$= \frac{12x^3}{4x} + \frac{8x^2}{4x} - \frac{4x}{4x}$
Divide:
$= 3x^2 + 2x - 1$
Answer: $3x^2 + 2x - 1$
Separate:
$= \frac{12x^3}{4x} + \frac{8x^2}{4x} - \frac{4x}{4x}$
Divide:
$= 3x^2 + 2x - 1$
Answer: $3x^2 + 2x - 1$
Example 2: $\frac{15x^4 - 10x^3 + 5x^2}{5x^2}$
$= \frac{15x^4}{5x^2} - \frac{10x^3}{5x^2} + \frac{5x^2}{5x^2}$
$= 3x^2 - 2x + 1$
Answer: $3x^2 - 2x + 1$
$= \frac{15x^4}{5x^2} - \frac{10x^3}{5x^2} + \frac{5x^2}{5x^2}$
$= 3x^2 - 2x + 1$
Answer: $3x^2 - 2x + 1$
Example 3: $\frac{24x^3y^2 - 18x^2y + 6xy}{6xy}$
$= 4x^2y - 3x + 1$
Answer: $4x^2y - 3x + 1$
$= 4x^2y - 3x + 1$
Answer: $4x^2y - 3x + 1$
14. Divide Polynomials Using Long Division
Polynomial Long Division: Similar to numerical long division
Dividend: The polynomial being divided
Divisor: The polynomial you're dividing by
Quotient: The answer
Remainder: What's left over (if any)
Dividend: The polynomial being divided
Divisor: The polynomial you're dividing by
Quotient: The answer
Remainder: What's left over (if any)
Steps for Polynomial Long Division:
Step 1: Arrange both polynomials in descending order of degree
Step 2: Divide leading term of dividend by leading term of divisor
Step 3: Multiply entire divisor by result from Step 2
Step 4: Subtract from dividend
Step 5: Bring down next term
Step 6: Repeat until degree of remainder < degree of divisor
Step 1: Arrange both polynomials in descending order of degree
Step 2: Divide leading term of dividend by leading term of divisor
Step 3: Multiply entire divisor by result from Step 2
Step 4: Subtract from dividend
Step 5: Bring down next term
Step 6: Repeat until degree of remainder < degree of divisor
Division Algorithm:
$$\text{Dividend} = (\text{Divisor})(\text{Quotient}) + \text{Remainder}$$
Or: $$\frac{\text{Dividend}}{\text{Divisor}} = \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$$
$$\text{Dividend} = (\text{Divisor})(\text{Quotient}) + \text{Remainder}$$
Or: $$\frac{\text{Dividend}}{\text{Divisor}} = \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$$
Example 1: Divide $(x^2 + 7x + 12) \div (x + 3)$
Step 1: $x^2 \div x = x$
Step 2: $x(x + 3) = x^2 + 3x$
Step 3: Subtract: $(x^2 + 7x) - (x^2 + 3x) = 4x$
Step 4: $4x \div x = 4$
Step 5: $4(x + 3) = 4x + 12$
Step 6: Subtract: $(4x + 12) - (4x + 12) = 0$
Answer: $x + 4$
x + 4
___________
x + 3 | x² + 7x + 12
x² + 3x
________
4x + 12
4x + 12
_______
0
Step 1: $x^2 \div x = x$
Step 2: $x(x + 3) = x^2 + 3x$
Step 3: Subtract: $(x^2 + 7x) - (x^2 + 3x) = 4x$
Step 4: $4x \div x = 4$
Step 5: $4(x + 3) = 4x + 12$
Step 6: Subtract: $(4x + 12) - (4x + 12) = 0$
Answer: $x + 4$
Example 2: Divide $(2x^3 - 5x^2 + 3x - 2) \div (x - 2)$
Answer: $2x^2 - x + 1$
2x² - x + 1
_________________
x - 2 | 2x³ - 5x² + 3x - 2
2x³ - 4x²
__________
-x² + 3x
-x² + 2x
_________
x - 2
x - 2
_____
0
Answer: $2x^2 - x + 1$
Example 3: Divide $(x^3 + 2x^2 - 5x + 7) \div (x + 3)$
Quotient: $x^2 - x - 2$
Remainder: $13$
Answer: $x^2 - x - 2 + \frac{13}{x + 3}$
Quotient: $x^2 - x - 2$
Remainder: $13$
Answer: $x^2 - x - 2 + \frac{13}{x + 3}$
Important Notes:
• If remainder is 0, divisor is a factor of dividend
• Always write answer in standard form
• Include placeholder terms (0x²) if needed
• Check: (Divisor)(Quotient) + Remainder = Dividend
• If remainder is 0, divisor is a factor of dividend
• Always write answer in standard form
• Include placeholder terms (0x²) if needed
• Check: (Divisor)(Quotient) + Remainder = Dividend
Summary: Polynomial Operations
| Operation | Rule | Example |
|---|---|---|
| Addition | Combine like terms | $(3x + 5) + (2x - 1) = 5x + 4$ |
| Subtraction | Distribute negative, then combine | $(5x - 2) - (3x + 4) = 2x - 6$ |
| Multiply by Monomial | Distribute to each term | $3x(x + 4) = 3x^2 + 12x$ |
| Multiply Binomials | Use FOIL | $(x + 2)(x + 3) = x^2 + 5x + 6$ |
| Divide by Monomial | Divide each term | $\frac{6x^2 + 4x}{2x} = 3x + 2$ |
| Long Division | Similar to numerical division | $(x^2 + 5x + 6) \div (x + 2) = x + 3$ |
Special Products Quick Reference
| Pattern Name | Formula | Example |
|---|---|---|
| Square of Sum | $(a + b)^2 = a^2 + 2ab + b^2$ | $(x + 3)^2 = x^2 + 6x + 9$ |
| Square of Difference | $(a - b)^2 = a^2 - 2ab + b^2$ | $(x - 5)^2 = x^2 - 10x + 25$ |
| Difference of Squares | $(a + b)(a - b) = a^2 - b^2$ | $(x + 4)(x - 4) = x^2 - 16$ |
Success Tips for Polynomials:
✓ Always combine LIKE TERMS only (same variable, same exponent)
✓ When subtracting, distribute the negative sign to ALL terms
✓ For multiplication, multiply EVERY term by EVERY term
✓ Remember FOIL: First, Outer, Inner, Last
✓ Memorize special product patterns to save time
✓ Write polynomials in standard form (highest to lowest degree)
✓ Check division by multiplying: (Divisor)(Quotient) + Remainder = Dividend
✓ Use algebra tiles for visual understanding
✓ Practice identifying the degree and type of polynomial
✓ Always simplify your final answer completely
✓ Always combine LIKE TERMS only (same variable, same exponent)
✓ When subtracting, distribute the negative sign to ALL terms
✓ For multiplication, multiply EVERY term by EVERY term
✓ Remember FOIL: First, Outer, Inner, Last
✓ Memorize special product patterns to save time
✓ Write polynomials in standard form (highest to lowest degree)
✓ Check division by multiplying: (Divisor)(Quotient) + Remainder = Dividend
✓ Use algebra tiles for visual understanding
✓ Practice identifying the degree and type of polynomial
✓ Always simplify your final answer completely