Math Problem Statement

solve

Find all x-intercepts and y-intercepts of the graph of the function f(x) = 2x^3 + 12x^2 + 16x.

Solution

To solve for the xx- and yy-intercepts of the function:

f(x)=2x3+12x2+16xf(x) = 2x^3 + 12x^2 + 16x

we’ll analyze the function at key points where xx and yy values intercept the axes.

Step 1: Finding the yy-intercept

The yy-intercept occurs when x=0x = 0. Substitute x=0x = 0 into f(x)f(x):

f(0)=2(0)3+12(0)2+16(0)=0f(0) = 2(0)^3 + 12(0)^2 + 16(0) = 0

So, the yy-intercept is at (0,0)(0, 0).

Step 2: Finding the xx-intercepts

The xx-intercepts occur where f(x)=0f(x) = 0. Set f(x)=0f(x) = 0 and solve for xx:

2x3+12x2+16x=02x^3 + 12x^2 + 16x = 0

Factor out 2x2x:

2x(x2+6x+8)=02x(x^2 + 6x + 8) = 0

This gives us two factors:

  1. 2x=02x = 0 which implies x=0x = 0.
  2. x2+6x+8=0x^2 + 6x + 8 = 0, a quadratic equation.

To solve x2+6x+8=0x^2 + 6x + 8 = 0, factor it further:

x2+6x+8=(x+4)(x+2)=0x^2 + 6x + 8 = (x + 4)(x + 2) = 0

Setting each factor equal to zero:

  • x+4=0x + 4 = 0 gives x=4x = -4.
  • x+2=0x + 2 = 0 gives x=2x = -2.

Summary

The intercepts are:

  • xx-intercepts: (4,0),(2,0),(0,0)(-4, 0), (-2, 0), (0, 0)
  • yy-intercept: (0,0)(0, 0)

Please enter the values:

  • xx-intercepts: 4,2,0-4, -2, 0
  • yy-intercept: 00

Do you want further details on any of the steps?

Here are five related questions that could help deepen understanding:

  1. How would you find the intercepts if the polynomial were in a different form?
  2. Can a polynomial function have more than one yy-intercept?
  3. How would the graph look for this polynomial, and what do the intercepts tell us about it?
  4. What is the multiplicity of each root for f(x)=2x3+12x2+16xf(x) = 2x^3 + 12x^2 + 16x?
  5. How do you find intercepts for more complex functions involving higher-degree terms?

Tip: Always start by factoring the polynomial completely to identify all roots for xx-intercepts efficiently.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Functions
Factoring

Formulas

y-intercept at x = 0
Solving for x-intercepts by setting f(x) = 0

Theorems

Zero Product Property

Suitable Grade Level

Grades 10-12