Factor each polynomial and explain how you decided which method to use. a) x^2+10 x+25 b) x^2-10

Factor each polynomial and explain how you decided which...

Key Concepts

Perfect Square Trinomials

This concept applies to quadratics where the expression can be written as the square of a binomial. It involves recognizing a pattern in the coefficients such that the middle term is twice the product of the square roots of the first and last terms. For example, in an expression like x² + 10x + 25, noticing that 10 is twice 5 and 25 is the square of 5 leads to factoring it as (x + 5)².

Difference of Squares

This method is used when a quadratic expression is in the form of one square minus another, which factors into the product of a sum and a difference. The general identity is a² - b² = (a - b)(a + b). This technique is especially useful when factoring expressions such as x² - 25, where 25 can be seen as 5², resulting in factors of (x - 5)(x + 5).

General Quadratic Factoring Techniques

This broader concept involves methods for factoring any quadratic expression of the form ax² + bx + c. The process typically includes finding two numbers that multiply to ac and add to b, which is sometimes referred to as splitting the middle term. This approach can be used when the quadratic does not immediately fit into a more specific pattern like a perfect square or a difference of squares.

Sum of Squares

A sum of squares refers to an expression like x² + 25 that appears to be the sum of two squared terms. Unlike the difference of squares, the sum of squares does not factor into real linear factors. While it can be factored over the complex numbers using the identity a² + b² = (a + bi)(a - bi), in the context of real numbers it is considered irreducible.

Transcript

00:01 Polynomial x square a x square plus 10x plus 25 we can see that this is following the type a square plus 2ab plus b square and which is a perfect square that is a plus b whole square so here in place of a we have x that is x square in place of 2ab we have 2 x x into 5 so in place of v we need 5 this is 5 squared that is 25 so this expression can be written as xx square plus 2 into x into 5 plus 5 square which is nothing but x plus 5 whole square which is x plus 5 into x plus 5 also next b this is also same this is x squared minus 10x plus 25 where in place of 2ab we have minus 2 ab so for minus 2 ab we can write this as first this is x square minus 2 into x into p plus b square which is x square which is x minus 5 whole square or you can say x minus 5 into x minus 5 for next c we have x square plus 26x plus 25 so here we can't we can see that this can't be a perfect square it is not following that pattern now we will factorize it we will factorize it using general method so this is a into c so constant of x square is 1 and constant term is 25 the product of this is 25 now we have to find those two numbers whose product is 25 but sum of those two numbers are 26 so we can say that 1 and 20 is the 2 number whose sum is 26 and product is 25 so we can replace 26 x with 1x plus 25 x which is 26 x so in this expression we'll replace 26 x x by x by x plus 25 so it can be written as x squared plus x plus 25x plus 25 now in first two terms we will take x common so we have x plus 1 in last two terms we will take 25 common so this is 25 into x plus 1 now in both the terms x plus 1 is common so it can be written as x plus 1 and remaining term is x plus 25 so for the next d which is x squared minus 25 so this can be written as this is just difference of squares so this is x square minus 5 square so this can be written as x minus 5 into x plus 5.

02:44 And for next e, which is x squared plus 25, so this is not following any of this pattern...

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