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Answer:

There are six problems circled in red.

As requested, only the analytical solutions are shown.

Question 16. Factor the polynomial

  • [tex]44+15h+h^2[/tex]

  • Answer: (h + 4) (h + 11)

Question 17. Factor the polynomial

  • [tex]40-22x+x^2[/tex]

  • Answer: (x - 2) (x -20)

Question 18. Factor the polynomial

  • [tex]-24-10x+x^2[/tex]

  • Answer: (x - 12) (x + 2)

Question 19. Factor the polynomial

  • [tex]-42-m+m^2[/tex]

  • Answer: (m - 7) (m + 6)

Question 20. Solve the equation

  • [tex]x^2-7x+12=0[/tex]

  • Answer: x = 4 and x = 3

Question 22. Solve the equation

  • [tex]x^2-6x=27[/tex]

  • Answer: x = 9 and x = - 3.

Explanation:

Question 16. Factor the polynomial

[tex]44+15h+h^2[/tex]

1. Rearrange the terms:

  • [tex]h^2+15h+44[/tex]

2. Open two parenthesis and place the common variable, h, as the first term of each.

  • [tex](h+a)(h+b)[/tex]

You are going to determine the values of a and b.

3. The values of a and b must satisfy that their product is 44 and their sum is 15.

  • a + b = 15 and a × b = 44

  • 11 and 4 meet those conditions: 11 + 4 = 15, and 11 × 4 = 44.

So, the factored polynomial is:

  • [tex](h+4)(h+11)[/tex]

Question 17. Factor the polynomial

[tex]40-22x+x^2[/tex]

1. Rearrange the terms:

  • [tex]x^2-22x+40[/tex]

2. Write two binomials; each with first term x

  • (x-a)(x-b)

The negative sign for a comes from the negative sign before 22x; the negative sign of b comes from the product of the signs of -22x and + 40.

3. Find a and b such that a + b = 22 and a × b = 40

  • 20 + 2 = 22 and 20 × 2 = 40, so the binomials are (x - 2) and (x - 20)

And the factored polynomial is:

  • [tex](x-2)(x-20)[/tex]

Question 18. Factor the polynomial

[tex]-24-10x+x^2[/tex]

1. Rearrange the terms

  • [tex]x^2-10x-24[/tex]

2. Sketch two binomials

  • [tex](x-a)(x+b)[/tex]

The negative sign for a corresponds to the negative sign for -10x, and the positive sign for b corresponds to the multplication of the signs of -10x and -4.

3. Find a and b such that their product is -24 and their aum is - 10.

Those numbers are - 12 and + 2:

  • - 12 + 2 = - 10
  • - 12 × (2) = - 24

Thus, the two factors are (x - 12) and (x +2).

And the factored polynomial is:

  • [tex](x-12)(x+2)[/tex]

Question 19. Factor the polynomial

[tex]-42-m+m^2[/tex]

1. Rearrange the terms

  • [tex]m^2-m-42[/tex]

2. Sketch the factors:

  • [tex](m-a)(m+b)[/tex]

The negative sign for a corresponds to the negative sign in front of m, the positive sign for b corresponds to the product of the signs of -m and -42.

3. Find two numbers, a and b, such that - a + b = - 1, and a × b = - 42

Those numbers are 6 and 7:

  • - 7 + 6 = - 1
  • -7 × 6 = - 42

Then, the factors are m - 7 and m + 6.

The factored polynomial is:

  • [tex](m-7)(m+6)[/tex]

Question 20. Solve the equation

[tex]x^2-7x+12=0[/tex]

1. Factor the left side:

Find two numbers that sum -7 and their product is + 12. Those two numbers are - 4 and - 3.

  • [tex](x-4)(x-3)=0[/tex]

2. Use the zero product rule

  • x - 4 = 0       or     x - 3 = 0

  • x = 4             or     x = 3

Both, x = 4 and x = 3 are solutions.

Hence, the solutions are x = 4 and x = 3.

Question 22. Solve the equation

[tex]x^2-6x=27[/tex]

1. Move 27 to the left side (using subtraction property of equalities):

  • [tex]x^2-6x-27=0[/tex]

2. Factor the polynomial.

Find two numbers whose sum is - 6 and their product is  - 27.

Those numbers are - 9 and + 3:

  • - 9 + 3 = - 6
  • - 9 × 3 = 27

  • [tex](x-9)(x+3)=0[/tex]

3. Zero product rule

[tex]x-9=0\\ \\ x=9\\ \\ x+3=0\\ \\ x=-3[/tex]

The two solutions are: x = 9 and x = -3

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