Question

1. express as a trinomial: $(2x - 8)(x + 9)$
2. express as a trinomial: $(2x - 3)(2x - 2)$
3. express as a trinomial: $(3x + 10)(3x + 3)$
4. express as a trinomial: $(2x - 9)(x - 6)$
5. express as a trinomial: $(2x + 4)(2x + 4)$
6. express as a trinomial: $(2x - 7)(3x + 10)$
7. express as a trinomial: $(x - 6)(x - 6)$
8. express as a trinomial: $(x + 7)(2x + 3)$
9. express $(x + 8)^2$ as a trinomial in standard form.
10. express $(x - 2)^2$ as a trinomial in standard form.
11. express $(x - 5)^2$ as a trinomial in standard form.
12. express $(x + 10)^2$ as a trinomial in standard form.
13. express $(x - 4)^2$ as a trinomial in standard form.

Answer:

Let's solve these problems one by one. We'll use the distributive property (FOIL method for binomials) to expand each expression and then combine like terms to get a trinomial.

Problem 18: Express \((2x - 8)(x + 9)\) as a trinomial

Step 1: Apply the FOIL method

First terms: \(2x \cdot x = 2x^2\)
Outer terms: \(2x \cdot 9 = 18x\)
Inner terms: \(-8 \cdot x = -8x\)
Last terms: \(-8 \cdot 9 = -72\)
So, \((2x - 8)(x + 9) = 2x^2 + 18x - 8x - 72\)

Step 2: Combine like terms

\(18x - 8x = 10x\)
So, the trinomial is \(2x^2 + 10x - 72\)

Problem 19: Express \((2x - 3)(2x - 2)\) as a trinomial

Step 1: Apply the FOIL method

First terms: \(2x \cdot 2x = 4x^2\)
Outer terms: \(2x \cdot (-2) = -4x\)
Inner terms: \(-3 \cdot 2x = -6x\)
Last terms: \(-3 \cdot (-2) = 6\)
So, \((2x - 3)(2x - 2) = 4x^2 - 4x - 6x + 6\)

Step 2: Combine like terms

\(-4x - 6x = -10x\)
So, the trinomial is \(4x^2 - 10x + 6\)

Problem 20: Express \((3x + 10)(3x + 3)\) as a trinomial

Step 1: Apply the FOIL method

First terms: \(3x \cdot 3x = 9x^2\)
Outer terms: \(3x \cdot 3 = 9x\)
Inner terms: \(10 \cdot 3x = 30x\)
Last terms: \(10 \cdot 3 = 30\)
So, \((3x + 10)(3x + 3) = 9x^2 + 9x + 30x + 30\)

Step 2: Combine like terms

\(9x + 30x = 39x\)
So, the trinomial is \(9x^2 + 39x + 30\)

Problem 21: Express \((2x - 9)(x - 6)\) as a trinomial

Step 1: Apply the FOIL method

First terms: \(2x \cdot x = 2x^2\)
Outer terms: \(2x \cdot (-6) = -12x\)
Inner terms: \(-9 \cdot x = -9x\)
Last terms: \(-9 \cdot (-6) = 54\)
So, \((2x - 9)(x - 6) = 2x^2 - 12x - 9x + 54\)

Step 2: Combine like terms

\(-12x - 9x = -21x\)
So, the trinomial is \(2x^2 - 21x + 54\)

Problem 22: Express \((2x + 4)(2x + 4)\) as a trinomial

Step 1: Apply the FOIL method

First terms: \(2x \cdot 2x = 4x^2\)
Outer terms: \(2x \cdot 4 = 8x\)
Inner terms: \(4 \cdot 2x = 8x\)
Last terms: \(4 \cdot 4 = 16\)
So, \((2x + 4)(2x + 4) = 4x^2 + 8x + 8x + 16\)

Step 2: Combine like terms

\(8x + 8x = 16x\)
So, the trinomial is \(4x^2 + 16x + 16\)

Problem 23: Express \((2x - 7)(3x + 10)\) as a trinomial

Step 1: Apply the FOIL method

First terms: \(2x \cdot 3x = 6x^2\)
Outer terms: \(2x \cdot 10 = 20x\)
Inner terms: \(-7 \cdot 3x = -21x\)
Last terms: \(-7 \cdot 10 = -70\)
So, \((2x - 7)(3x + 10) = 6x^2 + 20x - 21x - 70\)

Step 2: Combine like terms

\(20x - 21x = -x\)
So, the trinomial is \(6x^2 - x - 70\)

Problem 24: Express \((x - 6)(x - 6)\) as a trinomial

Step 1: Apply the FOIL method

First terms: \(x \cdot x = x^2\)
Outer terms: \(x \cdot (-6) = -6x\)
Inner terms: \(-6 \cdot x = -6x\)
Last terms: \(-6 \cdot (-6) = 36\)
So, \((x - 6)(x - 6) = x^2 - 6x - 6x + 36\)

Step 2: Combine like terms

\(-6x - 6x = -12x\)
So, the trinomial is \(x^2 - 12x + 36\)

Problem 25: Express \((x + 7)(2x + 3)\) as a trinomial

Step 1: Apply the FOIL method

First terms: \(x \cdot 2x = 2x^2\)
Outer terms: \(x \cdot 3 = 3x\)
Inner terms: \(7 \cdot 2x = 14x\)
Last terms: \(7 \cdot 3 = 21\)
So, \((x + 7)(2x + 3) = 2x^2 + 3x + 14x + 21\)

Step 2: Combine like terms

\(3x + 14x = 17x\)
So, the trinomial is \(2x^2 + 17x + 21\)

Problem 26: Express \((x + 8)^2\) as a trinomial in standard form

Recall that \((a + b)^2 = a^2 + 2ab + b^2\)
Here, \(a = x\) and \(b = 8\)

Step 1: Apply the formula

\((x + 8)^2 = x^2 + 2 \cdot x \cdot 8 + 8^2\)
\(= x^2 + 16x + 64\)

Problem 27: Express \((x - 2)^2\) as a trinomial in standard form

Recall that \((a - b)^2 = a^2 - 2ab + b^2\)
Here, \(a = x\) and \(b = 2\)

Step 1: Apply the formula

\((x - 2)^2 = x^2 - 2 \cdot x \cdot 2 + 2^2\)
\(= x^2 - 4x + 4\)

Problem 28: Express \((x - 5)^2\) as a trinomial in standard form

Recall that \((a - b)^2 = a^2 - 2ab + b^2\)
Here, \(a = x\) and \(b = 5\)

Step 1: Apply the formula

\((x - 5)^2 = x^2 - 2 \cdot x \cdot 5 + 5^2\)
\(= x^2 - 10x + 25\)

Problem 29: Express \((x + 10)^2\) as a trinomial in standard form

Recall that \((a + b)^2 = a^2 + 2ab + b^2\)
Here, \(a = x\) and \(b = 10\)

Step 1: Apply the formula

\((x + 10)^2 = x^2 + 2 \cdot x \cdot 10 + 10^2\)
\(= x^2 + 20x + 100\)

Problem 30: Express \((x - 4)^2\) as a trinomial in standard form

Recall that \((a - b)^2 = a^2 - 2ab + b^2\)
Here, \(a = x\) and \(b = 4\)

Step 1: Apply the formula

\((x - 4)^2 = x^2 - 2 \cdot x \cdot 4 + 4^2\)
\(= x^2 - 8x + 16\)

Final Answers:

1. \(2x^2 + 10x - 72\)
2. \(4x^2 - 10x + 6\)
3. \(9x^2 + 39x + 30\)
4. \(2x^2 - 21x + 54\)
5. \(4x^2 + 16x + 16\)
6. \(6x^2 - x - 70\)
7. \(x^2 - 12x + 36\)
8. \(2x^2 + 17x + 21\)
9. \(x^2 + 16x + 64\)
10. \(x^2 - 4x + 4\)
11. \(x^2 - 10x + 25\)
12. \(x^2 + 20x + 100\)
13. \(x^2 - 8x + 16\)