Factoring Polynomials Procedure



Factoring Polynomials Procedure

Given any polynomial in x of form a1xn + a2xn-1 + a3xn-2 +...+ an-1x + an, follow the following guidelines for factoring.

1. If the polynomial has a common monomial factor (that is, if all the coefficients are multiples of some number and/or if all terms in the polynomial contain the same variable), then factor out that monomial by writing it down and writing down the quotient of the original polynomial divided by the common factor.

Example: 8x4 + 16x3 – 12x2 + 20x = (4x)(2x3 + 4x2 – 3x + 5)

2. If the degree of the polynomial is greater than 2, then use synthetic division and the rational zeros theorem to discover any real rational roots. Reminder: if a power of x is missing from the expression, be sure to include a zero placeholder for that power of x.

Example: 4x4 - 24x3 + 35x2 + 6x - 9

p-term is 9 therefore p-factors are 1, 3, 9

q-term is 4 therefore q-factors are 1, 2, 4

Thus, possible rational zeros are the ratios of each p-factor to each q-factor.

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Synthetic Division:

prz

3 4 -24 35 6 -9

12 -36 -3 9

4 -12 -1 3 0

The rightmost column of the bottom line yields the remainder of the division, which happens to be the value of the polynomial at the possible rational zero (prz). If that value happens to be zero (as it is above), then we have discovered the location of a zero of the polynomial--a place where the graph of the polynomial crosses the x-axis. The rest of the bottom line consists of the coefficients of the terms in the quotient. Continuing synthetic division on this quotient will eventually allow you to factor the polynomial as completely as possible. This particular polynomial factors to

(x - 3)² (2x - 1) (2x + 1)

Note that x = 3 is a double root because of the squaring exponent, and the graph “bounces” off the x-axis at that point resembling a parabola in the immediate region of the zero.

3. If the polynomial is a cubic or quartic (degree of 3 or 4), try to group the terms of the polynomial in pairs or triples so that they contain a common binomial factor that may be extracted.

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Example:

4. If the polynomial is quadratic, you may extract squares, complete the square, factor into binomials, or use the quadratic formula to find the roots

Example of completing the

square:

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Example of extracting squares:

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Examples of Binomial factoring:

x² + 5x - 6 x² + x - 6

(x + 6)(x - 1) (x + 3)(x - 2)

x² + 7x + 6 x² - 5x + 6

(x + 6)(x + 1) (x - 2)(x - 3)

x² - 5x – 6 2x² + 7x - 15

(x - 6)(x + 1) (2x - 3)(x + 5)

Example of using the Quadratic Formula:

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Examples of Special Products factoring:

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Examples of Binomial factoring using Master Products method.

1) Find the product of the a and c terms, then list the factors of that product.

2) If the sign of the c term is positive, look for factors whose sum equals the b term; if the sign of the c term is negative, look for factors whose difference equals the b term.

3) Replace the b term in the expression with the combination of factors determined in step 2) and the appropriate signs to produce the sign of the b term.

4) Solve by grouping.

Example:

a b c

12x² - 23x - 24 Since the factors of 12 and 24 are quite numerous, the combinations you would have to try to find a difference of 23 could be quite time consuming. So the master products and grouping method is more efficient; indeed, it simplifies any factorable quadratic in four steps.

12 * 24 = 288. Factor pairs are (2,144) (3,96) (4,72) (6,48) (8,36) (9,32) (12,24) (16,18)

Since the sign of the c term is negative, we look for a factor pair whose difference is 23, and we will assign a negative sign to the larger factor since the b term is negative. We select (9,-32).

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Note: Always factor out the sign of the coefficient of the third term in line 2. The result will always be a common binomial factor which can be extracted in the next step.

5. Calculator Methods:

1. Using a graphing calculator's solve( function allows one to find the roots directly.

2. Alternatively, one can key a polynomial into a Y-variable (say Y1), set another Y-variable (say Y2) equal to zero, and then use the intersection option from the Calc menu to find the points of intersection for the polynomial (Y1) and the x-axis (Y2)— which will also be the roots of the polynomial.

3. An adept calculator addict could also use the Trace and ZOOM functions to discover the x-intercepts by successive approximation; however, the solutions generally will not be as precise as those obtained by methods 1 or 2.

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