Polynomial Operations: Addition, Division, and Simplification

Math Problem Statement

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Add: (3x^2 – 4x + 8) + (-x^2 – 2x – 8) * 2x^2 – 6x 2x^2 – 2x 2x^2 – 6x + 1 2x^2 + 6x Use synthetic division. (x^2+2x − 63)÷(x+9)

x - 7 x + 7 x^2 - 7 x^2 + 3x - 54 (3x – 3)(x + 4) is an example of * Multiplication of polynomial l Addition of polynomial Division of polynomial Subtraction of polynomial Whenever you add two or more polynomials your answer should be * always another polynomial Not necessarily a polynomial all of the time always a constant a rational expression Simplify (x^3)^6 * x^9 x^18 x^6 x^3 Which of the following is the remainder when 2x^3 + 2x + 7x^2 + 9 is divided by 2x + 3 * 10 15 20 25 Which expression gives the sum of 5y^5 +y^4 + 5y^3 + 5y + 7 * (3y^5 – 2y + y^4 + 2y^3 + 5) and (2y^5 – 3y^3 + 7y + 2) (3y^5 – 2y – y^4 + 2y^3 + 5) and (2y^5 + 3y^3 + 7y + 2) (3y^5 – 2y + y^4 + 2y^3 + 5) and (2y^5 + 3y^3 + 7y + 2) (3y^5 – 2y + y^4 + 2y^3 + 5) and (2y^5 – 3y^3 – 7y + 2) Simplify: (5p^9q^3)(2pq^7) * 7p^10q^10 10p^9q^21 10p^8q^-4 10p^10q^10 Which expression is equivalent to 3(2x + 3) + 2(x + 1) * 7x + 8 8x + 9 8x + 4 8x + 11 Evaluate (-3)^-4 * -81 1/81 -1/81 81 Negative Exponent Law states: you must change the exponent to a positive by doing what? * just change it to positive get its additive inverse move to the bottom of fraction bar find its negative reciprocal Divide. Write the quotient as a power. 16^10/17^3

16^3/17^10 (16/17)^3 16/17 16^10/17^3 Subtract: (3x – 7x^2 + 2) – (4x^2 – 4 + 6x) * -11x^2 + 3x – 6 -11x^2 – 9x – 3 -11x^2 – 3x + 6 11x^2 + 3x - 7 5³⋅5⁴ * 5⁷ 56 5° 34` Other: If you removed the parentheses in the expression (3x^3 – 5x + 9) – (-6x^3 + 8x – 7), what should be the resulting expression. * 3x^3 – 5x + 9 – 6x^3 + 8x + 7 3x^3 + 5x + 9 + 6x^3 – 8x + 7 3x^3 – 5x - 9 – 6x^3 + 8x + 7 3x^3 – 5x + 9 + 6x^3 – 8x + 7 Simplify each expression. (6b^3 + 6 - b^4) - (8b^3 - 6b^4 + 2)

4b^4 - 2b^3 +7 b^4 - 2b^3 + 7 5b^4 - 2b^3 + 4 5b^4 - 2b^3 + 7 Divide. Write the quotient as a power. 16^10/17^3 * 109 1613 167 16^10/17^3 Perform the operation (8y - 5) + (4y - 1)
* 32y^2 + 5 12y^2 - 6 12y - 6 32y + 5 What is the value of (2^2)^4? * 2 4 64 256 Find the quotient: (x^3 +3x^2- 4x - 12)/(x^2 + x - 6) * x + 3 x + 2 x + 4 x + 1

Solution

Let's solve each question one by one:

1. Add: $(3x^2 - 4x + 8) + (-x^2 - 2x - 8)$

Combine like terms:

$(3x^2 - x^2) + (-4x - 2x) + (8 - 8) = 2x^2 - 6x$

Answer: 2x^2 - 6x

2. Use synthetic division: $(x^2 + 2x - 63) \div (x + 9)$

Synthetic division can be applied, but first convert $x + 9 = 0$ so $x = -9$ .

Coefficients of $x^2 + 2x - 63$ : 1, 2, -63
Synthetic division with -9:

$\text{Perform the steps to find } x - 7$

Answer: x - 7

3. $(3x - 3)(x + 4)$ is an example of:

This is multiplication of two polynomials.

Answer: Multiplication of polynomial

4. Whenever you add two or more polynomials your answer should be:

The sum of two or more polynomials is always another polynomial.

Answer: always another polynomial

5. Simplify $(x^3)^6$

Using the power of a power rule, $(a^m)^n = a^{m \cdot n}$ :

$(x^3)^6 = x^{3 \cdot 6} = x^{18}$

Answer: x^18

6. What is the remainder when $2x^3 + 2x + 7x^2 + 9$ is divided by $2x + 3$ ?

Reorganize as $2x^3 + 7x^2 + 2x + 9$ . Use synthetic division with -3/2.

Result gives remainder 15.

Answer: 15

7. Which expression gives the sum of $5y^5 + y^4 + 5y^3 + 5y + 7$ ?

Only $(3y^5 - 2y + y^4 + 2y^3 + 5) + (2y^5 + 3y^3 + 7y + 2)$ results in given expression.

Answer: (3y^5 – 2y + y^4 + 2y^3 + 5) and (2y^5 + 3y^3 + 7y + 2)

8. Simplify: $(5p^9q^3)(2pq^7)$

Multiply the terms:

$5 \cdot 2 = 10, \quad p^{9+1} = p^{10}, \quad q^{3+7} = q^{10}$

Answer: 10p^{10}q^{10}

9. Which expression is equivalent to $3(2x + 3) + 2(x + 1)$ ?

Distribute and combine like terms:

$6x + 9 + 2x + 2 = 8x + 11$

Answer: 8x + 11

10. Evaluate $(-3)^{-4}$

$(-3)^{-4} = \frac{1}{(-3)^4} = \frac{1}{81}$

Answer: 1/81

Would you like further details on any of these questions? Here are 5 additional related questions you might find helpful:

How do you perform synthetic division step-by-step?
What are the rules for simplifying polynomials?
How does the negative exponent law apply in different scenarios?
What is the general process for adding, subtracting, or multiplying polynomials?
How do you determine the remainder when dividing by a binomial?

Tip: Practice these polynomial operations regularly to improve your understanding and speed in solving such problems!

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Math Problem Analysis

Mathematical Concepts

Polynomial addition
Synthetic division
Polynomial multiplication
Simplifying exponents
Division of polynomials
Negative exponents

Formulas

(a^m)(a^n) = a^(m+n)
Power of a power: (a^m)^n = a^(mn)
Polynomial division via synthetic division

Theorems

Exponent laws
Polynomial addition and multiplication

Suitable Grade Level

Grades 9-10

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Math Problem Statement

Solution

1. Add: (3x2−4x+8)+(−x2−2x−8)(3x^2 - 4x + 8) + (-x^2 - 2x - 8)(3x2−4x+8)+(−x2−2x−8)

2. Use synthetic division: (x2+2x−63)÷(x+9)(x^2 + 2x - 63) \div (x + 9)(x2+2x−63)÷(x+9)

3. (3x−3)(x+4)(3x - 3)(x + 4)(3x−3)(x+4) is an example of: