ಅಪವರ್ತಿಸಿ: (i) x³−2x²−x+2 (iii) x³+13x²+32x+20

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To factorize the expression x³ + 13x² + 32x + 20, we can use the Factor Theorem and polynomial division. Step 1: Use the Factor Theorem to find a root. We test integer divisors of the constant term (20). Let P(x) = x³ + 13x² + 32x + 20. Try x = −1: P(−1) = (−1)³ + 13(−1)² + 32(−1) + 20; P(−1) = −1 + 13(1) − 32 + 20; P(−1) = −1 + 13 − 32 + 20; P(−1) = 33 − 33 = 0; Since P(−1) = 0, (x − (−1)), which is (x + 1), is a factor of the polynomial. Step 2: Perform synthetic division to divide x³ + 13x² + 32x + 20 by (x + 1). -1 | 1 13 32 20 | -1 -12 -20 ------------------ 1 12 20 0 The quotient is x² + 12x + 20. Step 3: Factor the quadratic quotient x² + 12x + 20. We look for two numbers that multiply to 20 and add up to 12. These numbers are 2 and 10. x² + 12x + 20 = (x + 2)(x + 10) Step 4: Combine all factors to get the fully factorized expression. x³ + 13x² + 32x + 20 = (x + 1)(x + 2)(x + 10) The fully factorized expression is: (x + 1)(x + 2)(x + 10)

ಅಪವರ್ತಿಸಿ: (i) x³-2x²-x+2 (iii) x³+13x²+32x+20

Step-by-step solution

Use the Factor Theorem to find a root.

Perform synthetic division to divide $x^3 + 13x^2 + 32x + 20$ by $(x + 1)$ .

Factor the quadratic quotient $x^2 + 12x + 20$ .

Combine all factors to get the fully factorized expression.

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