Unit 5 Lesson 3 Practice 1. The polynomial function \( B(x)=x^{3}-21 x+20 \) has a known factor of \( (x-4) \). Rewrite \( B(x) \) as a product of linear factors using long division.

To rewrite the polynomial \( B(x)=x^{3}-21x+20 \) as a product of linear factors, we start by using long division to divide \( B(x) \) by the known factor \( (x-4) \). 1. Divide the leading term: \( x^3 \div x = x^2 \). 2. Multiply \( (x-4) \) by \( x^2 \) to get \( x^3 - 4x^2 \). 3. Subtract this from \( B(x) \): \[ B(x) - (x^3 - 4x^2) = 4x^2 - 21x + 20 \]. 4. Bring down the next term to get the new polynomial \( 4x^2 - 21x + 20 \). 5. Divide \( 4x^2 \div x = 4x \). 6. Multiply \( (x-4) \) by \( 4x \) to get \( 4x^2 - 16x \). 7. Subtract again: \[ (4x^2 - 21x + 20) - (4x^2 - 16x) = -5x + 20 \]. 8. Now divide \( -5x \div x = -5 \). 9. Multiply \( (x-4) \) by \( -5 \) to get \( -5x + 20 \). 10. Subtract to obtain the remainder: \[ (-5x + 20) - (-5x + 20) = 0 \]. Now, the division yields: \[ B(x) = (x - 4)(x^2 + 4x - 5). \] Next, we can factor \( x^2 + 4x - 5 \): - Find two numbers that multiply to \(-5\) and add to \(4\). These numbers are \(5\) and \(-1\). - Thus, we can factor it as \( (x + 5)(x - 1) \). Putting it all together, we have: \[ B(x) = (x - 4)(x + 5)(x - 1). \] So, the final answer is: \[ B(x) = (x - 4)(x + 5)(x - 1). \]