Add and subtract the polynomials and simplify: 7. (5x 2 + 6x-2)+(3x-2x+ 4) 8. (4×2-7x+1)-(x+9x-2) 9. (3x 2 + 2x + 8) + (9×2+7x -1) 10. (2x 2-5x -3) -(6×2-4x-8) 11. (7x-3) +(5x-4) 12. (2x+ 6)-(8x+9) Multiply the polynomials and simplify: 13. (7x 3-89(9/ + 5x-x) 14. (5x 3) (6×2+ 2x-9) 15. (10x+ 7) (6x-8) 16. (9x+ 4) (3×2-1) 17.(8x 3-2×2+ 3x) (Bx 2+ 7x-5) Solve each quadratic equation through factoring: 18, x 2 + 4x + 4 = 0 19. x2-14x+49 0 20. y2+ 7y-60 0 21. ?(x-3)-18 22, x 2 = 12x-36 23. 5x 2
The Correct Answer and Explanation is :
Let’s tackle the polynomial operations (addition, subtraction, and multiplication) one at a time, followed by solving the quadratic equations through factoring.
Addition and Subtraction of Polynomials:
- (5x² + 6x – 2) + (3x² – 2x + 4)
Combine like terms:
[
(5x^2 + 3x^2) + (6x – 2x) + (-2 + 4) = 8x^2 + 4x + 2
] - (4x² – 7x + 1) – (x² + 9x – 2)
Distribute the negative sign and combine like terms:
[
(4x^2 – x^2) + (-7x – 9x) + (1 + 2) = 3x^2 – 16x + 3
] - (3x² + 2x + 8) + (9x² + 7x – 1)
Combine like terms:
[
(3x^2 + 9x^2) + (2x + 7x) + (8 – 1) = 12x^2 + 9x + 7
] - (2x² – 5x – 3) – (6x² – 4x – 8)
Distribute the negative sign and combine like terms:
[
(2x^2 – 6x^2) + (-5x + 4x) + (-3 + 8) = -4x^2 – x + 5
] - (7x – 3) + (5x – 4)
Combine like terms:
[
(7x + 5x) + (-3 – 4) = 12x – 7
] - (2x + 6) – (8x + 9)
Distribute the negative sign and combine like terms:
[
(2x – 8x) + (6 – 9) = -6x – 3
]
Multiplication of Polynomials:
- (7x³ – 8x² + 5x – x) (This polynomial has an unclear component, assumed corrected)
Simply simplify the polynomial:
[
7x^3 – 8x^2 + 4x
] - (5x³)(6x² + 2x – 9)
Distribute (5x^3) across each term:
[
5x^3 \cdot 6x^2 + 5x^3 \cdot 2x – 5x^3 \cdot 9 = 30x^5 + 10x^4 – 45x^3
] - (10x + 7)(6x – 8)
Use the FOIL method:
[
(10x \cdot 6x) + (10x \cdot -8) + (7 \cdot 6x) – (7 \cdot 8) = 60x^2 – 80x + 42x – 56 = 60x^2 – 38x – 56
] - (9x + 4)(3x² – 1)
Distribute each term:
[
(9x \cdot 3x^2) + (9x \cdot -1) + (4 \cdot 3x^2) – (4 \cdot 1) = 27x^3 – 9x + 12x^2 – 4
] - (8x³ – 2x² + 3x)(8x² + 7x – 5)
Multiply each term by each term:
[
(8x^3 \cdot 8x^2) + (8x^3 \cdot 7x) – (8x^3 \cdot 5) – (2x^2 \cdot 8x^2) – (2x^2 \cdot 7x) + (2x^2 \cdot 5) + (3x \cdot 8x^2) + (3x \cdot 7x) – (3x \cdot 5) = 64x^5 + 56x^4 – 40x^3 – 16x^4 – 14x^3 + 10x^2 + 24x^3 + 21x^2 – 15x
]
Combine like terms:
[
64x^5 + 40x^4 + 10x^3 + 31x^2 – 15x
]
Solving Quadratic Equations by Factoring:
- (x^2 + 4x + 4 = 0)
Factor:
[
(x + 2)^2 = 0 \implies x = -2
] - (x^2 – 14x + 49 = 0)
Factor:
[
(x – 7)^2 = 0 \implies x = 7
] - (y^2 + 7y – 60 = 0)
Factor:
[
(y + 12)(y – 5) = 0 \implies y = -12, y = 5
] - (x^2 = 12x – 36)
Rewrite and factor:
[
x^2 – 12x + 36 = 0 \implies (x – 6)^2 = 0 \implies x = 6
] - (5x^2 = 0)
Simply solve:
[
x = 0
]
These operations and factoring procedures demonstrate basic polynomial manipulation techniques, including like term combination, distribution, FOIL, and straightforward factoring for quadratic equations. Each operation helps to clarify the mechanics of algebraic manipulation in solving and simplifying expressions.