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The Humongous Book of Algebra Problems by W. Michael Kelley

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Chapter Eleven — Polynomials
The Humongous Book of Algebra Problems
246
11.26 Calculate the product and simplify: (xy)(x
2
+ 6xy – 9y
2
).
Distribute each term of the left binomial through the right trinomial.
11.27 Calculate the product and simplify: (x + 3y – 1)(2xy + 7).
Distribute each term of the left trinomial through the right trinomial.
11.28 Calculate the product and simplify: (a – 3b + 2c)(4ab + 5c).
Distribute each term of the left trinomial through the right trinomial.
Long Division of Polynomials
A lot like long dividing integers
11.29 Calculate (x + 4) ÷ x using long division.
Rewrite the expression as a long division problem, with the divisor outside the
division symbol and the dividend within it.
Even though
both polynomials
have three terms,
it doesn’t change
the way you multiply.
Distribute x, then 3y,
and then –1 through
the right-hand
polynomial and add
everything together:
x(2x – y + 7) + 3y(2x
– y + 7) – 1(2x – y
+ 7).
Before you
try to perform
long division on
polynomials, make
sure you remember
how the process
works with whole
numbers—ip back to
Problems 2.3 and 2.5.
The two techniques
are basically the same,
except with whole
numbers you divide
one DIGIT at a time,
and with polynomials
you divide one
TERM at a time.
The divisor is
what youre dividing BY and
the dividend is what youre
dividing INTO.
Chapter Eleven — Polynomials
The Humongous Book of Algebra Problems
247
Divide the leftmost term of the dividend by the divisor: x ÷ x = 1. Write this
number above its like term in the dividend.
Multiply the newly placed number by the divisor: 1 · x = x. Write the opposite of
the result beneath the dividend, once again lining up like terms.
Combine like terms: xx = 0.
Add the next term of the dividend (+4).
Combine like terms: 0 + 4 = 4.
The degree of the divisor x is greater than the degree of the polynomial below
the horizontal line, so the process of long division is complete.
The number above the division symbol (1) is the quotient and the number
below the horizontal line (4) is the remainder. Write the solution using the
format below.
The number
4 in the dividend
x + 4 has the same
variable as 1 (because
neither of them have
any variables). That
means 1 and 4 are like
terms. When you write
numbers above the
division symbol, line
them up with their
like terms.
Whenever
you multiply a
number on top of the
division symbol by the
number out front, write
the OPPOSITE of that
number (in this case –x
instead of x) beneath
the dividend.
x = x
1
, so x has
degree one. 4 is a
constant and doesnt
have any variables,
so it has degree zero.
Because 1 > 0, youre
done dividing.
Chapter Eleven — Polynomials
The Humongous Book of Algebra Problems
248
Note: Problems 11.30–11.31 refer to the quotient (x
2
+ 6x – 9) ÷ (x – 2).
11.30 Calculate the quotient and remainder using long division.
Express the quotient as a long division problem.
Divide the leftmost term of the dividend by the leftmost term of the divisor:
x
2
÷ x = x. Write the answer above the like term 6x in the dividend.
Multiply the newly placed x by each term of the divisor and write the opposites of
the products beneath the dividend.
Combine like terms (6x + 2x = 8x) and add the next term of the dividend (–9).
Divide the leftmost term below the horizontal line by the leftmost term of the
divisor: 8x ÷ x = 8. Write the answer above the division symbol, multiply both
terms of the divisor by 8, and write the opposites of the products below 8x – 9.
Therefore, .
In other words,
x (from the divisor)
times what equals x
2
(from the dividend)?
The answers x:
x
.
x = x
2
. If you write
the polynomials from
highest to lowest powers
of x, you’ll always be
dealing with the terms
on the left: “The left
term of the divisor
times what equals
the left term of the
dividend?

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