Question

practice
factor each polynomial over the set of real numbers. use the factors to sketch the polynomial.
1 ( f(x) = 25x^2 - 10x - 24 )
2 ( f(x) = x^3 - 4x^2 - 9x + 36 )
3 ( f(x) = x^4 - 25x^2 + 144 )

Explanation:

Problem 1: \( f(x) = 25x^2 - 10x - 24 \)

Step 1: Use AC method

For \( ax^2 + bx + c \), find \( ac = 25 \times (-24) = -600 \). Find two numbers that multiply to -600 and add to -10. The numbers are -30 and 20.
Rewrite the middle term: \( 25x^2 - 30x + 20x - 24 \)

Step 2: Group and factor

Group as \( (25x^2 - 30x) + (20x - 24) \)
Factor out GCF from each group: \( 5x(5x - 6) + 4(5x - 6) \)

Step 3: Factor out common binomial

Factor out \( (5x - 6) \): \( (5x - 6)(5x + 4) \)

Step 1: Factor by grouping

Group as \( (x^3 - 4x^2) + (-9x + 36) \)

Step 2: Factor out GCF from each group

Factor out \( x^2 \) from first group and -9 from second: \( x^2(x - 4) - 9(x - 4) \)

Step 3: Factor out common binomial

Factor out \( (x - 4) \): \( (x - 4)(x^2 - 9) \)

Step 4: Factor difference of squares

\( x^2 - 9 = (x - 3)(x + 3) \), so final factors: \( (x - 4)(x - 3)(x + 3) \)

Step 1: Let \( u = x^2 \), rewrite as quadratic

\( u^2 - 25u + 144 \)

Step 2: Factor the quadratic

Find two numbers that multiply to 144 and add to -25. The numbers are -9 and -16.
Factor: \( (u - 9)(u - 16) \)

Step 3: Substitute back \( u = x^2 \)

\( (x^2 - 9)(x^2 - 16) \)

Step 4: Factor difference of squares

\( x^2 - 9 = (x - 3)(x + 3) \) and \( x^2 - 16 = (x - 4)(x + 4) \), so final factors: \( (x - 3)(x + 3)(x - 4)(x + 4) \)

Answer:

\( (5x - 6)(5x + 4) \)

Problem 2: \( f(x) = x^3 - 4x^2 - 9x + 36 \)