How to pull out the greatest common factor of polynomials

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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Factoring quadratic equations with coefficients

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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Completing the square for quadratic polynomials

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).

Completing the square when the roots of the polynomial are complex

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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Factoring polynomials using grouping

In this lesson we’ll look at factoring a polynomial using a method called grouping. When you have a polynomial, sometimes you can use factoring by grouping to help you get the factored parts. It means you need to look for terms in the polynomial that have values and terms in common and then group those parts together.

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Factoring quadratic polynomials

Factoring of quadratic polynomials (second-degree polynomials) is done by “un-FOILing,” which means we start with the result of a FOIL problem and work backwards to find the two binomial factors.

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Th quadratic formula for finding the roots of a quadratic

The quadratic formula is another way to solve quadratics that we can’t easily factor. You can think of the quadratic formula as a short-cut for completing the square. In fact, it was discovered by completing the square.

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How to factor the sum of two cubes

We’ll know when we have a sum of cubes because we’ll have two perfect cubes separated by addition. When that’s the case, we can take the cube (third) root of each term and use a formula to factor the sum of the cubes.

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Factoring the difference of two perfect squares

To factor a difference of two squares, you take the expressions that are squared and put both of them (in the given order) into the terms in both factors of the given binomial. In one factor, you’ll add the second expression to the first one; in the other factor, you’ll subtract the second expression from the first one.

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How to factor the difference of cubes

In this lesson we’ll look at how to recognize a difference of two cubes and then use a formula to factor it. We know we’re dealing with the difference of cubes when we have two perfect cubes separated by subtraction.

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Factoring out the greatest common factor from trinomials

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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Solving limits with factoring

If you tried to solve the limit with substitution and it didn’t work, factoring should be the next thing you try. The goal will be to factor the function, and then cancel any removable discontinuities, in order to simplify the function, so that it can be evaluated.

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Factoring to find common denominators for rational expressions

To add rational expressions, you need to find a common denominator, just like when you add fractions in which the numerator and denominator are just numbers. The difference is that finding the common denominator of rational expressions can be more complicated because their denominators can include variables.

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Zero theorem for the roots of a polynomial function

The zero theorem allows you to solve for the roots of a polynomial function. Just factor the polynomial, set it equal to 0, and solve for the variable to find the roots.

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