Polynomial factorization #12 - Questions and Answers

Question 1 Factor each polynomial 3x² - 10x + 8 Answer Step 1 Factor means to find two or more values whose products equal the original value. factorization is a simplifying algebraic expression by extracting a common factor. a polynomial is an expression in algebra that consists of two or more terms. polynomial equation an equation f(x)=0 where, f(x) is a polynomial. Quadratic equation degree is 2 if factoring the quadratic equation find the two numbers that add up to middle number and multiply together to get the last term and split the middle term by that two numbers and take out the common factor in first and last terms and equate the product to zero. Step 2 To factor the polynomial 3x² - 10x + 8 product the coefficient of the first term and the last constant term. that is 24 find out the two numbers which multiply together to get 24 and add up to get -10 those numbers are -6 and -4 split the middle term by -6 and -4. in the first term take out the common factor 3x and in the last term take out the common factor -4 and take the common factor (x-2). 3x²-10x+8=0 3x²-6x - 4x + 8 = 0 3x(x-2)-4(x-2)=0 (x-2)(3x-4)=0 The factors are (x-2)(3x-4) Question 2 Factor each polynomial by removing the common monomial factor. x³ - x² + x Answer Step 1 The monomial factor is a factor which consist only single term. We call an expression binomial if it consist 2 terms and trinomial if it consists three terms. Here we get a trinomial expression by removing common monomial factor. Then we have to find the factors of the resulting trinomial. we know that the general form of a trinomial is ax² + bx + c = 0 Step 2 We have a polynomial x³ - x² + x here we have to remove the common monomial factor. We can clearly identify that all the terms in the given polynomial x³ - x² + x is a multiple of x, therefor we can take x as a common monomial factor Thus we have x³ - x² + x = x(x² - x + 1) Now we have to find the factors of x² - x + 1 the trinomial. Now we have to find the factors of x² - x + 1 the trinomial. By trinomial factorization method, to factorize x² - x + 1 we have to look for two integers a and b such that sum of a and b equal to the the coefficient of x (-1) and the product of a and b equal to the constant number (1). We know that only two integers gives the product value 1 is 1 and 1 itself. and the same time we cannot have -1 as a sum of two this kind of numbers. Now we can use Discriminant to check the existence real roots or factors If the discriminant b² - 4ac>0, then we have real roots If the discriminant b² - 4ac<0, then we do not have real roots (we may get imaginary roots) We have trinomial x² - x + 1, where a=1, b=-1, c=1 b² - 4ac = (-1)² - 4(1)(1) =1-4 b²-4ac = -3 Thus we have discriminant b² - 4ac<0, hence we do not have any real roots or factors for x² - x + 1. Hence the factor of the polynomial x³ - x² + x is x(x² - x + 1). Question 3 Find the prime factorization of 147. Answer Step 1 Factor of 147: 1, 3, 7, 21, 49, 147 Step 2 So only 3 and 7 are prime numbers. Step 3 So prime factorization of 147 147 = 3 x 7 x 7 147 = 3 x 7² Question 4 What is the correct factorization of the expression 2x² + 7x - 15 Answer Step 1 Explanation of the answer is as follows Step 2 Solution: given that 2x² + 7x - 15 = 2x² + 10x - 3x - 15 = 2x(x+5)-3(x+5) = (x+5)(2x-3) Hence factor of 2x² + 7x - 15 is (x+5)(2x-3) Question 5 Factor completely 6a²b³ + 4a⁵b² - 12a⁶b⁴ Answer Step 1 Consider the provided expression: 6a²b³ + 4a⁵b² - 12a⁶b⁴ Step 2 Now, take out the least power of a that is a² from all the terms: a²(6b³ + 4a³b² - 12a⁴b⁴) Step 3 Now, take out the least power of b that is b² from all the terms: a²b²(6b + 4a³ - 12a⁴b²) Step 4 Now, further take the least common divisor (constant term) that is 2 from all the terms and get the required factorization: 2a²b²(3b + 2a³ - 6a⁴b²)