Task Sets: Identify zeros and graph polynomials - Common Core Math - HS Algebra

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Recall / Warm-Up

The polynomial $p(x) = (x - 3)(x + 7)(x - 1)$ has three zeros. Which list correctly identifies all three zeros?

$x = (x-3),\; x = (x+7),\; x = (x-1)$

$x = 3,\; x = -7,\; x = 1$

$x = -3,\; x = 7,\; x = -1$

$x = 3,\; x = 7,\; x = 1$

What is the zero of the factor $(x + 4)$ ?

$x = 4$

$x = -4$

$x = 0$

$x = 16$

Which pair of features determines the end behavior of a polynomial function?

The constant term and the number of zeros

The degree and the sign of the leading coefficient

The y-intercept and the total number of factors

The number of turning points and the constant term

Fluency Practice

Find the y-intercept of $p(x) = (x - 2)(x + 1)(x - 5)$ .

Enter the y-coordinate of the y-intercept.

What are the zeros of $p(x) = 2(x + 3)(x - 6)(x + 1)$ ?

$x = 3,\; x = -6,\; x = 1$

$x = -3,\; x = 6,\; x = -1$

$x = 2,\; x = 3,\; x = 6,\; x = 1$

$x = -3,\; x = 6,\; x = 1$

For $p(x) = (x + 2)^3(x - 1)^2$ , what is the multiplicity of the zero at $x = -2$ ?

$Two side-by-side graphs: left shows p(x) crossing through both x-intercepts; right shows q(x) bouncing at x=4 (even multiplicity) while still crossing at x=-1.$

For $p(x) = (x - 4)^2(x + 1)$ , what does the graph do at $x = 4$ ?

The graph crosses the x-axis at $x = 4$ because 4 is positive.

The graph crosses the x-axis at $x = 4$ because all zeros cause crossing.

The graph bounces (touches without crossing) at $x = 4$ because the multiplicity is 2 (even).

The graph has no x-intercept at $x = 4$ because $(x - 4)^2 \geq 0$ always.

Which statement correctly describes the zero $x = 2$ in $p(x) = (x - 2)^4(x + 3)$ ?

The graph crosses at $x = 2$ because the exponent 4 is greater than 1.

The graph bounces at $x = 2$ because the multiplicity is 4 (even).

The graph crosses at $x = 2$ because all polynomials cross their zeros.

The graph bounces at $x = 2$ because the zero is positive.

What is the end behavior of $p(x) = -3x^5 + 2x^3 - x + 7$ ?

As $x \to +\infty$ , $p(x) \to +\infty$ ; as $x \to -\infty$ , $p(x) \to -\infty$

As $x \to +\infty$ , $p(x) \to -\infty$ ; as $x \to -\infty$ , $p(x) \to +\infty$

Both ends go up ( $\uparrow\uparrow$ )

As $x \to +\infty$ , $p(x) \to +7$ ; as $x \to -\infty$ , $p(x) \to +7$

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Exercises: Identify Zeros and Graph Polynomials

Recall / Warm-Up

Fluency Practice