Question

Amanda was given the following polynomial and asked to factor completely. Find the errors that she made and explain how to fix her errors.

Answer 1

Given :-

• A polynomial 4x² + 22x + 24 .

To find:-

• The mistakes made by Amanda and how to fix the mistakes.

Answer:-

Given polynomial to us is ,

`\implies 4x^2 + 22x + 24 \\`

With respect to the standard form of polynomial ax² + bx + c , we have;

• a = 4 and c = 24

Step 1 :- Split the middle term into two parts p and q such that p × q = a × c and p + q = b

Such two numbers are 16 and 6 .

Hence, we can rewrite it as ,

`\implies 4x^2 + 16x + 6x + 24 \\`

But Amanda wrote
`4x^2+16x - 6x + 24` , so she made a mistake here by writing +6x as -6x .

Step 2:- Factor out the common terms .

`\implies 4x( x + 4) + 6(x+4) \\`

Step 3 :- Take out
`x+4` as common.

`\implies (x+4)(4x+6) \\`

This is our required factored form .

Hence we can see that Amanda made a mistake only in splitting the term term .

and we are done!

Answer 2

The error in Amanda's calculation is in line 2, where she incorrectly subtracted 6x instead of adding 6x, as well as incorrectly subtracting 24 instead of adding 24. Amanda also omitted the addition sign in line 3 of her calculations.

Explanation:

To factor a quadratic polynomial in the form ax² + bx + c, find two numbers that multiply to ac and sum to b.

Therefore, for the given polynomial 4x² + 22x + 24:

⇒ ac = 4 · 24 = 96

⇒ b = 22

The factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96.

Therefore, the factor pair that sums to 22 is 16 and 6.

Rewrite b as the sum of these two numbers:

⇒ 4x² + 22x + 24

⇒ 4x² + (16 + 6)x + 24

⇒ 4x² + 16x + 6x + 24

Therefore, the error in Amanda's calculation is in line 2, where she incorrectly subtracted 6x instead of adding 6x, as well as incorrectly subtracting 24 instead of adding 24. Amanda also omitted the addition sign in line 3 of her calculations.

The correct calculation is as follows:

4x² + 22x + 24

⇒ 4x² + 16x + 6x + 24

⇒ 4x(x + 4) + 6(x + 4)

⇒ (4x + 6)(x + 4)