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The key is that all the terms of the polynomial need to share the factor being taken out. Any factor that’s shared by all the terms is called a common factor, and the factor that consists of everything which is shared by all of them is known as the greatest common factor. Factoring is “un-distributing,” which means that we do the opposite of distributing and take out (or “factor out”) the same factor from each term of the polynomial (and divide each term by that factor to get “what’s left” once it’s taken out).
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In this lesson we’ll look at methods for factoring quadratic equations with coefficients in front of the x^2 term (that are not 1 or 0). Factoring means you’re taking the parts of an expression and rewriting it as parts that are being multiplied together (the factors). Factoring a quadratic equation means we will write equations of the form ax^2+bx+c into the form (px+r)(qx+s), where a, b, c, p, q, and s are all real numbers and a≠1,0.
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Adding and subtracting like terms is also known as combining like terms. Like terms are variables that have the same letter and same exponent. But like terms can have different coefficients. In other words, think about 3x^4+2x^4 as something like “3 apples + 2 apples,” where “apples” represents the x^4 term. We can simplify polynomials by looking for like terms and combining them together.
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The zeroes of a single-variable polynomial are the values of that variable at which the polynomial is equal to 0. Completing the square is a method we can use to find the zeroes of a quadratic polynomial. Another way to say this is that completing the square is a method we can use to solve the corresponding quadratic equation (the equation that has the quadratic polynomial on one side and 0 on the other side).
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When the discriminant is negative, the roots of the quadratic equation are complex, meaning that they’re complex numbers that include the imaginary number i. When the roots of a polynomial equation that are real numbers are also called real zeroes of the corresponding polynomial. Similarly, the roots of a polynomial equation that are complex numbers are also called complex zeroes of the corresponding polynomial.
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Do you remember doing long division? Now you probably use a calculator for most division problems. We’ll have to remember all those long division skills so that we can divide polynomials. Think about dividing polynomials as long division, but with variables.
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FOIL is a way to help you remember to multiply each term in the first set of parentheses by each term in the second set of parentheses. FOIL stands for Firsts, Outsides, Insides, Lasts, which is the order of the four terms in the result of the multiplication; it also indicates which terms in the given binomials are multiplied to produce each term in the result.
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When adding and subtracting polynomials, you’re really just looking for like terms to combine. The largest exponent in a polynomial is called the degree of the polynomial. The term with the largest exponent is called the leading term, because the terms of a polynomial are usually written in descending order of their exponents.
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The key to factoring is that every term in the trinomial needs to share the factor being taken out. Any factor that’s shared by all the terms is called a common factor, and the factor that consists of everything which is shared by all of them is known as the greatest common factor.
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