Polynomial Zeros Practice Worksheet
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Sketching Polynomials: Finding Zeros
Finding Zeros Example
Given: \( f(x) = x^2 + 7x + 10 \)
Step 1: Replace \( f(x) \) with zero: \( 0 = x^2 + 7x + 10 \)
Step 2: If necessary, factor the other side of the equation: \( 0 = (x + 2)(x + 5) \)
Step 3: Apply Zero Product Property (set each factor equal to zero) and solve:
\( x + 2 = 0 \)
− 2 − 2
\( x = -2 \)
\( x + 5 = 0 \)
− 5 − 5
\( x = -5 \)
Directions: Find the zeros of the provided functions, then graph the functions on Desmos to verify the x-intercepts.
1.) \( f(x) = (x + 2)(x + 7) \)
2.) \( f(x) = (x + 3)(x + 4) \)
3.) \( f(x) = (x - 5)(x + 6) \)
4.) \( f(x) = (x - 4)(2x - 3) \)
5.) \( f(x) = -(x + 8)(x + 1) \)
6.) \( f(x) = -(x + 3)(x - 9) \)
7.) \( f(x) = (x - 1)(x - 4)(x + 3) \)
8.) \( f(x) = -(x + 2)(x - 5)(x + 6) \)
9.) \( f(x) = x(x + 5)(x - 5) \)
10.) \( f(x) = -x(x + 3)(x - 1)(x - 6) \)
11.) \( f(x) = x^2 + 10x + 21 \)
12.) \( f(x) = x^2 - 2x - 15 \)
13.) \( f(x) = x^2 + 2x - 24 \)
14.) \( f(x) = x^2 - 8x + 15 \)
Polynomial Zeros Answer Key
TEACHER ANSWER KEY & STEP-BY-STEP SOLUTIONS POLYNOMIAL ZEROS MASTERY
Sketching Polynomials: Finding Zeros
Desmos Connection:
The real zeros \( x = c \) correspond directly to the graph's x-intercepts at coordinates \( (c, 0) \). Note that a negative leading coefficient reflects the graph vertically but does not alter the location of the zeros.
1.) \( f(x) = (x + 2)(x + 7) \)
\( 0 = (x + 2)(x + 7) \)
\( x + 2 = 0 \implies x = -2 \)
\( x + 7 = 0 \implies x = -7 \)
Zeros: \( x = -2, \; -7 \)
2.) \( f(x) = (x + 3)(x + 4) \)
\( 0 = (x + 3)(x + 4) \)
\( x + 3 = 0 \implies x = -3 \)
\( x + 4 = 0 \implies x = -4 \)
Zeros: \( x = -3, \; -4 \)
3.) \( f(x) = (x - 5)(x + 6) \)
\( 0 = (x - 5)(x + 6) \)
\( x - 5 = 0 \implies x = 5 \)
\( x + 6 = 0 \implies x = -6 \)
Zeros: \( x = 5, \; -6 \)
4.) \( f(x) = (x - 4)(2x - 3) \)
\( 0 = (x - 4)(2x - 3) \)
\( x - 4 = 0 \implies x = 4 \)
\( 2x - 3 = 0 \implies 2x = 3 \implies x = \frac{3}{2} \)
Zeros: \( x = 4, \; \frac{3}{2} \) (or 1.5)
5.) \( f(x) = -(x + 8)(x + 1) \)
\( 0 = -(x + 8)(x + 1) \)
\( x + 8 = 0 \implies x = -8 \)
\( x + 1 = 0 \implies x = -1 \)
Zeros: \( x = -8, \; -1 \)
6.) \( f(x) = -(x + 3)(x - 9) \)
\( 0 = -(x + 3)(x - 9) \)
\( x + 3 = 0 \implies x = -3 \)
\( x - 9 = 0 \implies x = 9 \)
Zeros: \( x = -3, \; 9 \)
7.) \( f(x) = (x - 1)(x - 4)(x + 3) \)
\( x - 1 = 0 \implies x = 1 \)
\( x - 4 = 0 \implies x = 4 \)
\( x + 3 = 0 \implies x = -3 \)
Zeros: \( x = 1, \; 4, \; -3 \)
8.) \( f(x) = -(x + 2)(x - 5)(x + 6) \)
\( x + 2 = 0 \implies x = -2 \)
\( x - 5 = 0 \implies x = 5 \)
\( x + 6 = 0 \implies x = -6 \)
Zeros: \( x = -2, \; 5, \; -6 \)
9.) \( f(x) = x(x + 5)(x - 5) \)
\( x = 0 \)
\( x + 5 = 0 \implies x = -5 \)
\( x - 5 = 0 \implies x = 5 \)
Zeros: \( x = 0, \; -5, \; 5 \)
10.) \( f(x) = -x(x + 3)(x - 1)(x - 6) \)
\( -x = 0 \implies x = 0 \)
\( x + 3 = 0 \implies x = -3 \)
\( x - 1 = 0 \implies x = 1 \)
\( x - 6 = 0 \implies x = 6 \)
Zeros: \( x = 0, \; -3, \; 1, \; 6 \)
11.) \( f(x) = x^2 + 10x + 21 \)
\( 0 = (x + 3)(x + 7) \)
\( x + 3 = 0 \implies x = -3 \)
\( x + 7 = 0 \implies x = -7 \)
Zeros: \( x = -3, \; -7 \)
12.) \( f(x) = x^2 - 2x - 15 \)
\( 0 = (x - 5)(x + 3) \)
\( x - 5 = 0 \implies x = 5 \)
\( x + 3 = 0 \implies x = -3 \)
Zeros: \( x = 5, \; -3 \)
13.) \( f(x) = x^2 + 2x - 24 \)
\( 0 = (x + 6)(x - 4) \)
\( x + 6 = 0 \implies x = -6 \)
\( x - 4 = 0 \implies x = 4 \)
Zeros: \( x = -6, \; 4 \)
14.) \( f(x) = x^2 - 8x + 15 \)
\( 0 = (x - 3)(x - 5) \)
\( x - 3 = 0 \implies x = 3 \)
\( x - 5 = 0 \implies x = 5 \)
Zeros: \( x = 3, \; 5 \)
