Factoring Polynomials1 3x² - 2x - 52 10m² + 17m+63 2m² + 63m + 1454 3x⁷+36x⁵+ 108x³5 12v² - 4v - 166 3x² - 8x + 47 10u³- 130u² + 400u8 121a² - 66ab + 9b²9 4w² - 17w - 1510 2k² + 19k - 1011 2m² + 63m + 14512 15a²+45ab + 60ab²​

Step-by-step explanation:Let's factor each polynomial:1. **\(3x^2 - 2x - 5\)** To factor, find two numbers that multiply to \(3 \times (-5) = -15\) and add up to \(-2\). These numbers are \(-5\) and \(3\). \[ 3x^2 - 2x - 5 = 3x^2 - 5x + 3x - 5 \] Group and factor: \[ = (3x^2 - 5x) + (3x - 5) = x(3x - 5) + 1(3x - 5) = (x + 1)(3x - 5) \] So, the factored form is \((x + 1)(3x - 5)\).2. **\(10m^2 + 17m + 6\)** Find two numbers that multiply to \(10 \times 6 = 60\) and add up to \(17\). These numbers are \(12\) and \(5\). \[ 10m^2 + 17m + 6 = 10m^2 + 12m + 5m + 6 \] Group and factor: \[ = (10m^2 + 12m) + (5m + 6) = 2m(5m + 6) + 1(5m + 6) = (2m + 1)(5m + 6) \] So, the factored form is \((2m + 1)(5m + 6)\).3. **\(2m^2 + 63m + 145\)** Find two numbers that multiply to \(2 \times 145 = 290\) and add up to \(63\). These numbers are \(58\) and \(5\). \[ 2m^2 + 63m + 145 = 2m^2 + 58m + 5m + 145 \] Group and factor: \[ = (2m^2 + 58m) + (5m + 145) = 2m(m + 29) + 5(m + 29) = (2m + 5)(m + 29) \] So, the factored form is \((2m + 5)(m + 29)\).4. **\(3x^7 + 36x^5 + 108x^3\)** Factor out the greatest common factor \(3x^3\): \[ 3x^7 + 36x^5 + 108x^3 = 3x^3(x^4 + 12x^2 + 36) \] The quadratic \(x^4 + 12x^2 + 36\) can be factored as \((x^2 + 6)^2\): \[ 3x^3(x^2 + 6)^2 \] So, the factored form is \(3x^3(x^2 + 6)^2\).5. **\(12v^2 - 4v - 16\)** Find two numbers that multiply to \(12 \times (-16) = -192\) and add up to \(-4\). These numbers are \(-16\) and \(12\). \[ 12v^2 - 4v - 16 = 12v^2 - 16v + 12v - 16 \] Group and factor: \[ = (12v^2 - 16v) + (12v - 16) = 4v(3v - 4) + 4(3v - 4) = 4(v + 1)(3v - 4) \] So, the factored form is \(4(v + 1)(3v - 4)\).6. **\(3x^2 - 8x + 4\)** Find two numbers that multiply to \(3 \times 4 = 12\) and add up to \(-8\). These numbers are \(-6\) and \(-2\). \[ 3x^2 - 8x + 4 = 3x^2 - 6x - 2x + 4 \] Group and factor: \[ = (3x^2 - 6x) - (2x - 4) = 3x(x - 2) - 2(x - 2) = (3x - 2)(x - 2) \] So, the factored form is \((3x - 2)(x - 2)\).7. **\(10u^3 - 130u^2 + 400u\)** Factor out the greatest common factor \(10u\): \[ 10u^3 - 130u^2 + 400u = 10u(u^2 - 13u + 40) \] Factor \(u^2 - 13u + 40\): Find two numbers that multiply to \(40\) and add up to \(-13\). These numbers are \(-8\) and \(-5\): \[ u^2 - 13u + 40 = (u - 8)(u - 5) \] So, the factored form is \(10u(u - 8)(u - 5)\).8. **\(121a^2 - 66ab + 9b^2\)** Factor as a quadratic in \(a\): Find two numbers that multiply to \(121 \times 9 = 1089\) and add up to \(-66\). These numbers are \(-59\) and \(-7\): \[ 121a^2 - 66ab + 9b^2 = (11a - 3b)(11a - 3b) \] So, the factored form is \((11a - 3b)^2\).9. **\(4w^2 - 17w - 15\)** Find two numbers that multiply to \(4 \times (-15) = -60\) and add up to \(-17\). These numbers are \(-20\) and \(3\): \[ 4w^2 - 17w - 15 = 4w^2 - 20w + 3w - 15 \] Group and factor: \[ = (4w^2 - 20w) + (3w - 15) = 4w(w - 5) + 3(w - 5) = (4w + 3)(w - 5) \] So, the factored form is \((4w + 3)(w - 5)\).10. **\(2k^2 + 19k - 10\)** Find two numbers that multiply to \(2 \times (-10) = -20\) and add up to \(19\). These numbers are \(20\) and \(-1\): \[ 2k^2 + 19k - 10 = 2k^2 + 20k - k - 10 \] Group and factor: \[ = (2k^2 + 20k) - (k + 10) = 2k(k + 10) - 1(k + 10) = (2k - 1)(k + 10) \] So, the factored form is \((2k - 1)(k + 10)\).11. **Duplicate of problem 3** - Already factored as \((2m + 5)(m + 29)\).12. **\(15a^2 + 45ab + 60ab^2\)** This seems to be a typo; the correct polynomial should be \(15a^2 + 45ab + 60b^2\): Factor out the greatest common factor \(15\): \[ 15a^2 + 45ab + 60b^2 = 15(a^2 + 3ab + 4b^2) \] The expression \(a^2 + 3ab + 4b^2\) is not factorable further with simple integers, so: The factored form is \(15(a^2 + 3ab + 4b^2)\).

Answered by bautistaryza402 | 2024-09-10