Math Problem Statement

Factor the following functions fully: a) f(x) = x^3 + 9x^2 + 8x - 60 b) f(x) = x^3 - 7x - 6 c) f(x) = x^4 - 5x^2 + 4

Solution

Let's work through the factoring process for each of the given functions.

a) f(x)=x3+9x2+8x60f(x) = x^3 + 9x^2 + 8x - 60

Step 1: Try grouping terms.

  • Group the first two terms and the last two terms:
    (x3+9x2)+(8x60)(x^3 + 9x^2) + (8x - 60)

Step 2: Factor out the greatest common factor (GCF) from each group.

  • From x3+9x2x^3 + 9x^2, factor out x2x^2:
    x2(x+9)x^2(x + 9)
  • From 8x608x - 60, factor out 4:
    4(2x15)4(2x - 15)

The expression is now:
x2(x+9)+4(2x15)x^2(x + 9) + 4(2x - 15)

This does not factor further easily by grouping, so this is likely the final factored form for now.


b) f(x)=x37x6f(x) = x^3 - 7x - 6

Step 1: Use synthetic or trial and error for factoring. Try factoring using possible roots of the cubic equation.

Possible rational roots are factors of -6 (constant term) divided by factors of 1 (leading coefficient of x3x^3), i.e., ±1,±2,±3,±6\pm 1, \pm 2, \pm 3, \pm 6.

Testing x=1x = -1:

  • Substituting into the function:
    (1)37(1)6=1+76=0(-1)^3 - 7(-1) - 6 = -1 + 7 - 6 = 0

So, x=1x = -1 is a root.

Step 2: Use synthetic division to divide f(x)f(x) by x+1x + 1.

Performing the division of x37x6x^3 - 7x - 6 by x+1x + 1 gives x2x6x^2 - x - 6.

Step 3: Factor x2x6x^2 - x - 6.

  • x2x6=(x3)(x+2)x^2 - x - 6 = (x - 3)(x + 2)

Thus, the fully factored form is:
f(x)=(x+1)(x3)(x+2)f(x) = (x + 1)(x - 3)(x + 2)


c) f(x)=x45x2+4f(x) = x^4 - 5x^2 + 4

Step 1: Let y=x2y = x^2, so the equation becomes:
y25y+4y^2 - 5y + 4

Step 2: Factor the quadratic in yy.
y25y+4=(y4)(y1)y^2 - 5y + 4 = (y - 4)(y - 1)

Step 3: Substitute back y=x2y = x^2.

  • (x24)(x21)(x^2 - 4)(x^2 - 1)

Step 4: Factor the difference of squares.

  • x24=(x2)(x+2)x^2 - 4 = (x - 2)(x + 2)
  • x21=(x1)(x+1)x^2 - 1 = (x - 1)(x + 1)

Thus, the fully factored form is:
f(x)=(x2)(x+2)(x1)(x+1)f(x) = (x - 2)(x + 2)(x - 1)(x + 1)


Would you like any further details on these steps or any clarifications?

Additional Practice Questions:

  1. How do you apply synthetic division to cubic functions?
  2. What are the common techniques for factoring polynomials by grouping?
  3. Can you explain the process of factoring the difference of squares?
  4. How do you determine possible rational roots of a polynomial?
  5. What are the techniques to factor higher-degree polynomials, such as quartic ones?

Tip:

When factoring polynomials, always check for possible common factors and apply grouping or synthetic division techniques for higher-degree polynomials.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Factoring by Grouping
Synthetic Division
Difference of Squares

Formulas

Factoring cubic and quartic polynomials
Synthetic division for polynomials
Difference of squares: a^2 - b^2 = (a - b)(a + b)

Theorems

Rational Root Theorem
Factor Theorem

Suitable Grade Level

Grades 9-12