Mathematics exam 6 pptx

The FOIL Method

The FOIL method can be used when multiplying bino￾mials. FOIL stands for the order used to multiply the

terms: First, Outer,Inner, and Last. To multiply binomi￾als, you multiply according to the FOIL order and then

add the like terms of the products.

Example

(3x + 1)(7x + 10)

3x and 7x are the first pair of terms,

3x and 10 are the outermost pair of terms,

1 and 7x are the innermost pair of terms, and

1 and 10 are the last pair of terms.

Therefore, (3x)(7x) + (3x)(10) + (1)(7x) +

(1)(10) = 21x2 + 30x + 7x + 10.

After we combine like terms, we are left with the

answer: 21x2 + 37x + 10.

Factoring

Factoring is the reverse of multiplication:

2(x + y) = 2x + 2y Multiplication

2x + 2y = 2(x + y) Factoring

Three Basic Types of Factoring

1. Factoring out a common monomial.

10x2 − 5x = 5x(2x − 1) and

xy − zy = y(x − z)

2. Factoring a quadratic trinomial using the reverse

of FOIL:

y2 − y − 12 = (y − 4) (y + 3) and

z2 − 2z + 1 = (z − 1)(z − 1) = (z − 1)2

3. Factoring the difference between two perfect

squares using the rule:

a2 − b2 = (a + b)(a − b) and

x2 − 25 = (x + 5)(x − 5)

Removing a Common Factor

If a polynomial contains terms that have common fac￾tors, the polynomial can be factored by dividing by the

greatest common factor.

Example

In the binomial 49x3 + 21x, 7x is the greatest

common factor of both terms.

Therefore, you can divide 49x3 + 21x by 7x to

get the other factor.

49x3

7

+

x

21x =

4

7

9

x

x3

+

2

7

1

x

x

= 7x2 + 3

Thus, factoring 49x3 + 21x results in

7x(7x2 + 3).

Quadratic Equations

A quadratic equation is an equation in which the great￾est exponent of the variable is 2, as in x2 + 2x − 15 = 0. A

quadratic equation has two roots, which can be found by

breaking down the quadratic equation into two simple

equations.

Example

Solve x2 + 5x + 2x + 10 = 0.

x2 + 7x + 10 = 0 Combine like terms.

(x + 5)(x + 2) = 0 Factor.

x + 5 = 0 or x + 2 = 0

x

−

=

5 −

−

5 5 x

−

=

2 −

−

2 2

Now check the answers.

−5 + 5 = 0 and −2 + 2 = 0

Therefore, x is equal to both −5 and −2.

Inequalities

Linear inequalities are solved in much the same way as

simple equations. The most important difference is that

when an inequality is multiplied or divided by a negative

number, the inequality symbol changes direction.

Example

10 > 5 but if you multiply by −3,

(10) − 3 < (5)−3

−30 < −15

Solving Linear Inequalities

To solve a linear inequality, isolate the variable and solve

the same as you would in a first-degree equation.

Remember to reverse the direction of the inequality sign

if you divide or multiply both sides of the equation by a

negative number.

–ALGEBRA, FUNCTIONS, AND PATTERNS–

415