Multiplication of polynomials (Page 2/2)

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$(a + b) (c + d) = a c + a d + b c + b d$

This method is commonly called the FOIL method .

F First terms
O Outer terms
I Inner terms
L Last terms

$(a + b) (2 + 3) = \underset{2 terms}{\underset{︸}{(a + b) + (a + b)}} + \underset{3 terms}{\underset{︸}{(a + b) + (a + b) + (a + b)}}$

Rearranging,

$\begin{array}{l} = a + a + b + b + a + a + a + b + b + b \\ = 2 a + 2 b + 3 a + 3 b \end{array}$

Combining like terms,

$= 5 a + 5 b$

This use of the distributive property suggests the following rule.

Multiplying a polynomial by a polynomial

To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.

Sample set c

Perform the following multiplications and simplify.

Finding the product of 'a plus six' and 'a plus three' using the FOIL method. See the longdesc for a full description.

With some practice, the second and third terms can be combined mentally.

Finding the product of two binomials 'x plus y' and 'two x plus four y' using the FOIL method. See the longdesc for a full description.

Finding the product of two polynomials 'x squared plus four' and 'x squared plus seven x plus two' using the FOIL method. See the longdesc for a full description.

Finding the product of two binomials 'a minus four' and 'a minus three' using the FOIL method. See the longdesc for a full description.

$\begin{array}{l} {(m - 3)}^{2} & = & (m - 3) (m - 3) \\ = & m \cdot m + m (- 3) - 3 \cdot m - 3 (- 3) \\ = & m^{2} - 3 m - 3 m + 9 \\ = & m^{2} - 6 m + 9 \end{array}$

$\begin{array}{l} {(x + 5)}^{3} & = & (x + 5) (x + 5) (x + 5) & Associate the first two factors . \\ = & [(x + 5) (x + 5)] (x + 5) \\ = & [x^{2} + 5 x + 5 x + 25] (x + 5) \\ = & [x^{2} + 10 x + 25] (x + 5) \\ = & x^{2} \cdot x + x^{2} \cdot 5 + 10 x \cdot x + 10 x \cdot 5 + 25 \cdot x + 25 \cdot 5 \\ = & x^{3} + 5 x^{2} + 10 x^{2} + 50 x + 25 x + 125 \\ = & x^{3} + 15 x^{2} + 75 x + 125 \end{array}$

Practice set c

Find the following products and simplify.

$(a + 1) (a + 4)$

$a^{2} + 5 a + 4$

$(m - 9) (m - 2)$

$m^{2} - 11 m + 18$

$(2 x + 4) (x + 5)$

$2 x^{2} + 14 x + 20$

$(x + y) (2 x - 3 y)$

$2 x^{2} - x y - 3 y^{2}$

$(3 a^{2} - 1) (5 a^{2} + a)$

$15 a^{4} + 3 a^{3} - 5 a^{2} - a$

$(2 x^{2} y^{3} + x y^{2}) (5 x^{3} y^{2} + x^{2} y)$

$10 x^{5} y {}^{5}+ 7 x^{4} y^{4} + x^{3} y^{3}$

$(a + 3) (a^{2} + 3 a + 6)$

$a^{3} + 6 a^{2} + 15 a + 18$

$(a + 4) (a + 4)$

$a^{2} + 8 a + 16$

$(r - 7) (r - 7)$

$r^{2} - 14 r + 49$

${(x + 6)}^{2}$

$x^{2} + 12 x + 36$

${(y - 8)}^{2}$

$y^{2} - 16 y + 64$

Sample set d

Perform the following additions and subtractions.

$\begin{array}{l} 3 x + 7 + (x - 3) . & We must first remove the parentheses . They are preceded by \\ a " + " sign, so we remove them and leave the sign of each \\ term the same . \\ 3 x + 7 + x - 3 & Combine like terms . \\ 4 x + 4 \end{array}$

$\begin{array}{l} 5 y^{3} + 11 - (12 y^{3} - 2) . & We first remove the parentheses . They are preceded by a \\ "-" sign, so we remove them and change the sign of each \\ term inside them . \\ 5 y^{3} + 11 - 12 y^{3} + 2 & Combine like terms . \\ - 7 y^{3} + 13 \end{array}$

Add $4 x^{2} + 2 x - 8$ to $3 x^{2} - 7 x - 10$ .

$\begin{array}{l} (4 x^{2} + 2 x - 8) + (3 x^{2} - 7 x - 10) \\ 4 x^{2} + 2 x - 8 + 3 x^{2} - 7 x - 10 \\ 7 x^{2} - 5 x - 18 \end{array}$

Subtract $8 x^{2} - 5 x + 2$ from $3 x^{2} + x - 12$ .

$\begin{array}{l} (3 x^{2} + x - 12) - (8 x^{2} - 5 x + 2) \\ 3 x^{2} + x - 12 - 8 x^{2} + 5 x - 2 \\ - 5 x^{2} + 6 x - 14 \end{array}$

Be very careful not to write this problem as

$3 x^{2} + x - 12 - 8 x^{2} - 5 x + 2$

This form has us subtracting only the very first term, $8 x^{2}$ , rather than the entire expression. Use parentheses.
Another incorrect form is

$8 x^{2} - 5 x + 2 - (3 x^{2} + x - 12)$

This form has us performing the subtraction in the wrong order.

Practice set d

Perform the following additions and subtractions.

$6 y^{2} + 2 y - 1 + (5 y^{2} - 18)$

$11 y^{2} + 2 y - 19$

$(9 m - n) - (10 m + 12 n)$

$- m - 13 n$

Add $2 r^{2} + 4 r - 1$ to $3 r^{2} - r - 7$ .

$5 r^{2} + 3 r - 8$

Subtract $4 s - 3$ from $7 s + 8$ .

$3 s + 11$

Exercises

For the following problems, perform the multiplications and combine any like terms.

$7 (x + 6)$

$7 x + 42$

$4 (y + 3)$

$6 (y + 4)$

$6 y + 24$

$8 (m + 7)$

$5 (a - 6)$

$5 a - 30$

$2 (x - 10)$

$3 (4 x + 2)$

$12 x + 6$

$6 (3 x + 4)$

$9 (4 y - 3)$

$36 y - 27$

$5 (8 m - 6)$

$- 9 (a + 7)$

$- 9 a - 63$

$- 3 (b + 8)$

$- 4 (x + 2)$

$- 4 x - 8$

$- 6 (y + 7)$

$- 3 (a - 6)$

$- 3 a + 18$

$- 9 (k - 7)$

$- 5 (2 a + 1)$

$- 10 a - 5$

$- 7 (4 x + 2)$

$- 3 (10 y - 6)$

$- 30 y + 18$

$- 8 (4 y - 11)$

$x (x + 6)$

$x^{2} + 6 x$

$y (y + 7)$

$m (m - 4)$

$m^{2} - 4 m$

$k (k - 11)$

$3 x (x + 2)$

$3 x^{2} + 6 x$

$4 y (y + 7)$

$6 a (a - 5)$

$6 a^{2} - 30 a$

$9 x (x - 3)$

$3 x (5 x + 4)$

$15 x^{2} + 12 x$

$4 m (2 m + 7)$

$2 b (b - 1)$

$2 b^{2} - 2 b$

$7 a (a - 4)$

$3 x^{2} (5 x^{2} + 4)$

$15 x^{4} + 12 x^{2}$

$9 y^{3} (3 y^{2} + 2)$

$4 a^{4} (5 a^{3} + 3 a^{2} + 2 a)$

$20 a^{7} + 12 a^{6} + 8 a^{5}$

$2 x^{4} (6 x^{3} - 5 x^{2} - 2 x + 3)$

$- 5 x^{2} (x + 2)$

$- 5 x^{3} - 10 x^{2}$

$- 6 y^{3} (y + 5)$

$2 x^{2} y (3 x^{2} y^{2} - 6 x)$

$6 x^{4} y^{3} - 12 x^{3} y$

$8 a^{3} b^{2} c (2 a b^{3} + 3 b)$

$b^{5} x^{2} (2 b x - 11)$

$2 b^{6} x^{3} - 11 b^{5} x^{2}$

$4 x (3 x^{2} - 6 x + 10)$

$9 y^{3} (2 y^{4} - 3 y^{3} + 8 y^{2} + y - 6)$

$18 y^{7} - 27 y^{6} + 72 y^{5} + 9 y^{4} - 54 y^{3}$

$- a^{2} b^{3} (6 a b^{4} + 5 a b^{3} - 8 b^{2} + 7 b - 2)$

$(a + 4) (a + 2)$

$a^{2} + 6 a + 8$

$(x + 1) (x + 7)$

$(y + 6) (y - 3)$

$y^{2} + 3 y - 18$

$(t + 8) (t - 2)$

$(i - 3) (i + 5)$

$i^{2} + 2 i - 15$

$(x - y) (2 x + y)$

$(3 a - 1) (2 a - 6)$

$6 a^{2} - 20 a + 6$

$(5 a - 2) (6 a - 8)$

$(6 y + 11) (3 y + 10)$

$18 y^{2} + 93 y + 110$

$(2 t + 6) (3 t + 4)$

$(4 + x) (3 - x)$

$- x^{2} - x + 12$

$(6 + a) (4 + a)$

$(x^{2} + 2) (x + 1)$

$x^{3} + x^{2} + 2 x + 2$

$(x^{2} + 5) (x + 4)$

$(3 x^{2} - 5) (2 x^{2} + 1)$

$6 x^{4} - 7 x^{2} - 5$

$(4 a^{2} b^{3} - 2 a) (5 a^{2} b - 3 b)$

$(6 x^{3} y^{4} + 6 x) (2 x^{2} y^{3} + 5 y)$

$12 x^{5} y^{7} + 30 x^{3} y^{5} + 12 x^{3} y^{3} + 30 x y$

$5 (x - 7) (x - 3)$

$4 (a + 1) (a - 8)$

$4 a^{2} - 28 a - 32$

$a (a - 3) (a + 5)$

$x (x + 1) (x + 4)$

$x^{3} + 5 x^{2} + 4 x$

$x^{2} (x + 5) (x + 7)$

$y^{3} (y - 3) (y - 2)$

$y^{5} - 5 y^{4} + 6 y^{3}$

$2 a^{2} (a + 4) (a + 3)$

$5 y^{6} (y + 7) (y + 1)$

$5 y^{8} + 40 y^{7} + 35 y^{6}$

$a b^{2} (a^{2} - 2 b) (a + b^{4})$

$x^{3} y^{2} (5 x^{2} y^{2} - 3) (2 x y - 1)$

$10 x^{6} y^{5} - 5 x^{5} y^{4} - 6 x^{4} y^{3} + 3 x^{3} y^{2}$

$6 (a^{2} + 5 a + 3)$

$8 (c^{3} + 5 c + 11)$

$8 c^{3} + 40 c + 88$

$3 a^{2} (2 a^{3} - 10 a^{2} - 4 a + 9)$

$6 a^{3} b^{3} (4 a^{2} b^{6} + 7 a b^{8} + 2 b^{10} + 14)$

$24 a^{5} b^{9} + 42 a^{4} b^{11} + 12 a^{3} b^{13} + 18 a^{3} b^{3}$

$(a - 4) (a^{2} + a - 5)$

$(x - 7) (x^{2} + x - 3)$

$x^{3} - 6 x^{2} - 10 x + 21$

$(2 x + 1) (5 x^{3} + 6 x^{2} + 8)$

$(7 a^{2} + 2) (3 a^{5} - 4 a^{3} - a - 1)$

$21 a^{7} - 22 a^{5} - 15 a^{3} - 7 a^{2} - 2 a - 2$

$(x + y) (2 x^{2} + 3 x y + 5 y^{2})$

$(2 a + b) (5 a^{2} + 4 a^{2} b - b - 4)$

$10 a^{3} + 8 a^{3} b + 4 a^{2} b^{2} + 5 a^{2} b - b^{2} - 8 a - 4 b - 2 a b$

${(x + 3)}^{2}$

${(x + 1)}^{2}$

$x^{2} + 2 x + 1$

${(x - 5)}^{2}$

${(a + 2)}^{2}$

$a^{2} + 4 a + 4$

${(a - 9)}^{2}$

$- {(3 x - 5)}^{2}$

$- 9 x^{2} + 30 x - 25$

$- {(8 t + 7)}^{2}$

For the following problems, perform the indicated operations and combine like terms.

$3 x^{2} + 5 x - 2 + (4 x^{2} - 10 x - 5)$

$7 x^{2} - 5 x - 7$

$- 2 x^{3} + 4 x^{2} + 5 x - 8 + (x^{3} - 3 x^{2} - 11 x + 1)$

$- 5 x - 12 x y + 4 y^{2} + (- 7 x + 7 x y - 2 y^{2})$

$2 y^{2} - 5 x y - 12 x$

$(6 a^{2} - 3 a + 7) - 4 a^{2} + 2 a - 8$

$(5 x^{2} - 24 x - 15) + x^{2} - 9 x + 14$

$6 x^{2} - 33 x - 1$

$(3 x^{3} - 7 x^{2} + 2) + (x^{3} + 6)$

$(9 a^{2} b - 3 a b + 12 a b^{2}) + a b^{2} + 2 a b$

$9 a^{2} b + 13 a b^{2} - a b$

$6 x^{2} - 12 x + (4 x^{2} - 3 x - 1) + 4 x^{2} - 10 x - 4$

$5 a^{3} - 2 a - 26 + (4 a^{3} - 11 a^{2} + 2 a) - 7 a + 8 a^{3} + 20$

$17 a^{3} - 11 a^{2} - 7 a - 6$

$2 x y - 15 - (5 x y + 4)$

Add $4 x + 6$ to $8 x - 15$ .

$12 x - 9$

Add $5 y^{2} - 5 y + 1$ to $- 9 y^{2} + 4 y - 2$ .

Add $3 (x + 6)$ to $4 (x - 7)$ .

$7 x - 10$

Add $- 2 (x^{2} - 4)$ to $5 (x^{2} + 3 x - 1)$ .

Add four times $5 x + 2$ to three times $2 x - 1$ .

$26 x + 5$

Add five times $- 3 x + 2$ to seven times $4 x + 3$ .

Add $- 4$ times $9 x + 6$ to $- 2$ times $- 8 x - 3$ .

$- 20 x - 18$

Subtract $6 x^{2} - 10 x + 4$ from $3 x^{2} - 2 x + 5$ .

Substract $a^{2} - 16$ from $a^{2} - 16$ .

Exercises for review

( [link] ) Simplify ${(\frac{15 x^{2} y^{6}}{5 x y^{2}})}^{4}$ .

( [link] ) Express the number 198,000 using scientific notation.

$1.98 \times 10^{5}$

( [link] ) How many $4 a^{2} x^{3}' s$ are there in $- 16 a^{4} x^{5}$ ?

( [link] ) State the degree of the polynomial $4 x y^{3} + 3 x^{5} y - 5 x^{3} y^{3}$ , and write the numerical coefficient of each term.

$degree is 6; 4, 3, - 5$

( [link] ) Simplify $3 (4 x - 5) + 2 (5 x - 2) - (x - 3)$ .

Questions & Answers

what is biology

Hajah Reply

the study of living organisms and their interactions with one another and their environments

AI-Robot

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Victoria Reply

HOW CAN MAN ORGAN FUNCTION

Alfred Reply

the diagram of the digestive system

Assiatu Reply

allimentary cannel

Ogenrwot

How does twins formed

William Reply

They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together

Oluwatobi

what is genetics

Josephine Reply

Genetics is the study of heredity

Misack

how does twins formed?

Misack

What is manual

Hassan Reply

discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles

Joseph Reply

what is biology

Yousuf Reply

the study of living organisms and their interactions with one another and their environment.

Wine

discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form

Joseph Reply

what is the blood cells

Shaker Reply

list any five characteristics of the blood cells

Shaker

lack electricity and its more savely than electronic microscope because its naturally by using of light

Abdullahi Reply

advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear

Abdullahi

cell theory state that every organisms composed of one or more cell,cell is the basic unit of life

Abdullahi

is like gone fail us

DENG

cells is the basic structure and functions of all living things

Ramadan

What is classification

ISCONT Reply

is organisms that are similar into groups called tara

Yamosa

in what situation (s) would be the use of a scanning electron microscope be ideal and why?

Kenna Reply

A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,

Hilary

cell is the building block of life.

Condoleezza Reply

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Source: OpenStax, Algebra i for the community college. OpenStax CNX. Dec 19, 2014 Download for free at http://legacy.cnx.org/content/col11598/1.3

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