How to Factor Difficult Polynomials and Factoring
How to multiply and factorize 3 polynomials
A polynomial is an algebraic expression that contains one or more terms. The terms can only be added, subtracted or multiplied become. Multiplying polynomials is the process of combining all terms using the distributive property. This process combines all of the similar terms. Multiplying polynomials combines all of the terms, leaving the polynomial in its expanded form. Factoring polynomials is the reverse process of breaking down an expanded form into its simplest forms.
How to multiply three polynomials
Examine the expression (x + 6) (x - 9) (2x + 4 ^ 2). This statement reads: The amount of x plus six times the amount of x minus nine times the amount of two x plus four squared. Rewrite the problem so that you address the first two polynomials (x + 6) (x - 9). It's easier to multiply in small, manageable steps rather than trying to tackle the entire expression.
Multiply the two Polynomials with the FOIL method. F stands for the first two terms, O stands for the outer terms, I for the inner terms and L for the last terms. Some people prefer to draw arrows to keep their multiplication steps organized.
Multiply the first two terms, X x X = x ^ 2 or x square. Multiply the outer terms, X x (-9) = -9x. Multiply the inner terms, 6 x X = 6x. Multiply the last terms, 6 x (-9) = -54. So far, the multiplication answer should be x ^ 2 - 9x + 6x - 54.
Combine similar terms. You cannot combine the expression x ^ 2 with the other xs because the exponent can change the x factor drastically. Instead, combine the individual x's, -9x + 6x = -3x. The answer so far is x ^ 2 - 3x - 54.
Rewrite the polynomial to include the remaining polynomial. (x ^ 2 - 3x - 54) (2x + 4 ^ 2). First, do the math in parentheses, 4 ^ 2 = 16. Multiply by the FOIL process.
Multiply the first terms, x ^ 2 by 2x = 2x ^ 3. If you are multiplying exponents, multiply the base for a product of 2x, then add the exponents for the answer 2x ^ 3. Multiply the outer terms, x ^ 2, by 16 = 16x ^ 2. Multiply the inner term, -3x, by the first and last terms. -3x x 2x = -6x ^ 2 and -3x x 16 = -48x. Multiply the last term, -54, by the first and last term. -54 x 2x = -108x and -54 x 16 = -864. Your problem should be: 2x ^ 3 + 16x ^ 2 - 6x ^ 2 - 48x - 108x - 864.
Combine similar terms. 2x ^ 3 + 10x ^ 2 - 156x - 864.
How to Factor Three Polynomials
Examine the expression 5x ^ 2 + 35x + 30.
Look for the biggest common factor, in which case 5 goes into all three terms. Write five outside the brackets, 5 (...) (...). The inside is blank for now but will be filled in when the issue is considered.
Divide all three terms by the GFC, five. Factoring is the opposite of multiplying and takes expressions to their simplest forms. Five goes once in 5x ^ 2 and leaves only the x ^ 2. Five goes back seven times 35x and 7x. Five goes in 30, six times and leaves six behind. The answer so far is 5 (x ^ 2 + 7x + 6).
Factor out the brackets. First, look at the first and last terms in parentheses. Are they squares, so can the numbers be broken down into a simple square root? No they are not. X ^ 2 is obviously squared, but there is no square root of 6. So you'll have to use trial and error to find out the simplest form of the bracket.
Write a set of brackets, leave the inside blank for now. Don't forget the five from the first factorization step. 5 (...) (...). Well what do you need to make x a square? Another x. So put that in the brackets. 5 (x ...) (x ...). You can see that when you use FOIL, the first two terms are squared equal to x.
The terms for 6 are excluded. They are 6 x 1 = 6 and 2 x 3 = 6. But what factors should you use to make the middle terms, the I in FOIL, add up to 7? Look at the factors. Is 2 + 3 equal to 7? No, but 6 + 1 does. Write these factors in the brackets. 5 (x ... 6) (x ... 1).
Choose your character. Because both 7x and 6 are positive, your signs will both be positive. 5 (x + 6) (x + 1).
Multiply the brackets by FOIL to check your work. X x X = x ^ 2, X x 1 = x, 6 x X = 6x and 6 x 1 = 6. Combine like terms, X ^ 2 + 7x + 6, which is the same as the problem after the second factorization step.
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