Question

Suppose that \(f(x)=3 x^{3}+16 x^{2}+3 x-10 .\) Find the zeros of \(f(x+3)\).

Short answer

The zeros of \( f(x+3) \) are approximately -4, -4/3, and -14/3.

Step-by-step solution

Substitute Variable

Given the function, we need to substitute \( f(x+3) \). Begin by replacing every instance of x in \( f(x)\) with \( x+3 \).

Substitute in Polynomial

Substitute \( x+3 \) into the polynomial: \[ f(x+3) = 3(x+3)^{3} + 16(x+3)^{2} + 3(x+3) - 10 \]

Expand the Cubic Term

Expand \( (x+3)^{3} \), the cubic term: \[ (x+3)^{3} = (x+3)(x+3)(x+3) = x^3 + 9x^2 + 27x + 27 \]

Expand the Quadratic Term

Expand \( (x+3)^{2} \), the quadratic term: \[ (x+3)^{2} = (x+3)(x+3) = x^2 + 6x + 9 \]

Substitute Expanded Terms Back

Substitute the expanded terms back into the polynomial: \[ f(x+3) = 3(x^3 + 9x^2 + 27x + 27) + 16(x^2 + 6x + 9) + 3(x + 3) - 10 \]

Combine Like Terms

Combine the like terms: \[ f(x+3) = 3x^3 + 27x^2 + 81x + 81 + 16x^2 + 96x + 144 + 3x + 9 - 10 \] Simplify to get: \[ f(x+3) = 3x^3 + 43x^2 + 180x + 224 \]

Find the Zeros

Set the polynomial equal to zero to find the zeros: \[ 3x^3 + 43x^2 + 180x + 224 = 0 \]

Solve for x

Solve the cubic equation for \( x \) to find the zeros. Since solving a cubic equation analytically can be complex, it may be necessary to use numerical methods or graphing software to find the approximate roots: \( x = -4, -4/3, -14/3 \).

Key concepts

substitution method

The substitution method involves replacing a given variable or term with another expression to simplify the problem. In our exercise, we are asked to find the zeros of the polynomial function \( f(x+3) \). This means we need to replace every instance of \( x \) in \( f(x) \) with \( x+3 \).
So, starting with the original function, \( f(x) = 3x^3 + 16x^2 + 3x - 10 \), we substitute \( x+3 \) wherever \( x \) appears:
\[ f(x+3) = 3(x+3)^3 + 16(x+3)^2 + 3(x+3) - 10 \] By doing this substitution, we transform the original problem into a different one that still holds all the same properties of the function.

expanding polynomial functions

Expanding polynomial functions involves multiplying out expressions and combining like terms to form a single polynomial. For expanding, we will do it step-by-step.

Step 1: Expand the cubic term \( (x+3)^3 \):
\[ (x+3)^3 = (x+3)(x+3)(x+3) = x^3 + 9x^2 + 27x + 27 \]
Step 2: Expand the quadratic term \( (x+3)^2 \):
\[ (x+3)^2 = (x+3)(x+3) = x^2 + 6x + 9 \]
Step 3: Substitute these expanded terms back into the polynomial:
\[ f(x+3) = 3(x^3 + 9x^2 + 27x + 27) + 16(x^2 + 6x + 9) + 3(x + 3) - 10 \]
This step ensures the polynomial is in its expanded form, with all terms clearly displayed, without any parentheses.

solving cubic equations

Solving cubic equations can be more challenging than solving linear or quadratic equations due to their higher degree. Once we have the expanded polynomial in the form \( f(x+3) = 3x^3 + 43x^2 + 180x + 224 \), we set it equal to zero to find the zeros:
\[ 3x^3 + 43x^2 + 180x + 224 = 0 \] Cubic equations generally have three roots. To find these roots, one can:

• Use factorization methods if the polynomial can be factored.
• Apply numerical methods or graphing tools to approximate the roots.

For this exercise, the roots have been approximated as \( x = -4, -4/3, -14/3 \). These are the values of \( x \) for which the polynomial equals zero.

combining like terms

Combining like terms is a method to simplify polynomial expressions by ensuring all similar terms are added or subtracted to form single terms. After substituting and expanding, we have:\[ f(x+3) = 3(x^3 + 9x^2 + 27x + 27) + 16(x^2 + 6x + 9) + 3(x + 3) - 10 \] Next, we distribute and combine like terms:

• Cubic terms: There is only one cubic term, \( 3x^3 \).
• Quadratic terms: Combine \( 27x^2 \) and \( 16x^2 \) to get \( 43x^2 \).
• Linear terms: Combine \( 81x \), \( 96x \), and \( 3x \) to get \( 180x \).
• Constant terms: Combine \( 81 \), \( 144 \), \( 9 \), and \( -10 \) to get \( 224 \).

Thus, we have:
\[ f(x+3) = 3x^3 + 43x^2 + 180x + 224 \]. This cleaner form is crucial for further solving or analyzing the polynomial.