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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

Aislinn Reply

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Siyaka Reply

A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance

Jude Reply

Can you compute that for me. Ty

Jude

what is the dimension formula of energy?

David Reply

what is viscosity?

David

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emma Reply

what is chemistry

Youesf Reply

what is inorganic

emma

Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes

Adjei

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Adjanou

chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.

Pedro

A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity

Krampah Reply

2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

Sahid Reply

you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

Samuel Reply

can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

Joseph Reply

Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

Joseph

Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

Joseph

"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

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answer

Magreth

progressive wave

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Mujahid

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

yasuo 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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