Question

factor the polynomial.

\\10x^4 - 8x^3 + 20x^2 - 16x\\

\\10x^4 - 8x^3 + 20x^2 - 16x = \square\\
(factor completely.)

Explanation:

⚡ Using what you learned: Factoring by Grouping

Step 1: Factor out the greatest common factor (GCF)

Identify the GCF of all terms in the polynomial \( 10x^4 - 8x^3 + 20x^2 - 16x \).

The GCF of the coefficients \( 10, -8, 20, -16 \) is \( 2 \).
The GCF of the variable parts \( x^4, x^3, x^2, x \) is \( x \).

Factor out \( 2x \):

$$ 2x(5x^3 - 4x^2 + 10x - 8) $$

Step 2: Factor by grouping inside the parentheses

Group the terms inside the parentheses into two pairs:

$$ (5x^3 - 4x^2) + (10x - 8) $$

Factor out the GCF from each group:

• From \( 5x^3 - 4x^2 \), factor out \( x^2 \):

$$ x^2(5x - 4) $$

• From \( 10x - 8 \), factor out \( 2 \):

$$ 2(5x - 4) $$

Combine the grouped terms:

$$ (x^2 + 2)(5x - 4) $$

Step 3: Write the complete factored form

Include the GCF factored out in Step 1:

$$ 2x(x^2 + 2)(5x - 4) $$

Answer:

$$ 2x(x^2 + 2)(5x - 4) $$