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596 C H A P T E R 5 : A LG E B R A I C M E T H O D S
TEKS
AR.4D Determine the
linear factors of a polyno-
mial function of degree two
and of degree three when
represented symbolically
and tabularly and graphi-
cally where appropriate.
MATHEMATICAL
PROCESS SPOTLIGHT
AR.1C Select tools,
including real objects,
manipulatives, paper and
pencil, and technology
as appropriate, and tech-
niques, including mental
math, estimation, and num-
ber sense as appropriate, to
solve problems.
ELPS
2I Demonstrate
listening comprehension of
increasingly complex spo-
ken English by following
directions, retelling or sum-
marizing spoken messages,
responding to questions
and requests, collaborat-
ing with peers, and taking
notes commensurate with
content and grade-level
needs.
VOCABULARY
zeroes, linear, factors, poly-
nomial
MATERIALS
596 C H A P T E R 5 : A LG E B R A I C M E T H O D S
FOCUSING QUESTION How can graphs, tables, and symbolic representa-
tions be used to determine linear factors of polynomial functions?
LEARNING OUTCOMES
■ ■ I can determine the linear factors of polynomial functions from tables, graphs, or symbolic representations. ■ ■ I can select tools, including paper and pencil or technology, to solve problems.
Decomposing Polynomial
Functions
Joshua Tree National Park
ENGAGE
Which of the following functions have a linear factor of 2 x – 1? ■ ■ 2 x^2 – x ■ ■ 2 x^2 + 7 x – 4 ■ ■ 2 x^2 – x – 3 ■ ■ 4 x^3 – 10 x^2 + 4 x 2x^2 – x, 2x^2 + 7x – 4, and 4x^3 – 10x^2 + 4x
EXPLORE
Work with a partner. For each of the following pairs of polynomial functions, each partner should select one function. Write the polynomial function as a product of linear factors. Take turns explaining to your partner how you determined the linear factors. Select one of the questions from the Question Bank to ask your partner, listening carefully to the answer. Check your factors using graphing technology.
1. f(x) = (x + 3)(2x + 1)
g(x) = x(x + 1)(x – 2)
QUESTION BANK ✔ How did you determine the linear factors? ✔ From which representation is it easier for you to determine the linear factors? Explain your reasoning. ✔ How does the degree of the polynomial compare to the number of linear factors you could have?
x -3 -2.5 -2 -1.5 -1 -0. f ( x ) 0 -2 -3 -3 -2 0
x -3 -2 -1 0 1 2 g ( x ) -30 -8 0 0 -2 0
INTEGRATING TECHNOLOGY
Encourage students
to use technology to
graph functions given
in symbolic form. With
the graphing technology,
students can calculate the
zeroes of the functions and
then use those zeroes to
write linear factors.
For functions given in
table form, students can
use graphing technology
to create a scatterplot or
use finite differences to
determine whether the
function is quadratic
(degree two) or cubic
(degree three).
SUPPORTING ENGLISH LANGUAGE LEARNERS
As students are learning
the English language, they
need the opportunity to
demonstrate their listening
comprehension. One
way is to pair students
up and have them ask
questions and respond to
questions ( ELPS 2I ) that are
commensurate with content
and grade-level needs.
REFLECT ANSWERS:
Factoring from any representation relies on identifying zeroes of a function and then using those
zeros to determine linear factors that will make the function value equal to zero for a given x-value.
Using a table relies on the x-coordinate of the x-intercept, using a graph relies on a zero of the func-
tion, and using symbolic representations relies on the numerical relationships among the terms. If
there is a constant factor that vertically dilates the graph or reflects it across the x-axis, factoring by
tables and graphs may not reveal that constant factor. However, factoring from a symbolic represen-
tation reveals that constant factor as a common factor among all terms in the function.
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2. 3. f ( x ) = x^2 + 10 x + 21, g ( x ) = x^3 + 5 x^2 + 2 x – 8 f(x) = (x + 3)(x + 7) g(x) = (x – 1)(x + 2)(x + 4)
4.
f(x) = (x + 3)^2 g(x) = (x + 2)(x – 2)^2
5. t ( x ) = 4 x^2 + 3 x – 27 q(x) = (x + 4)(2x – 3)
p(x) = (5x + 2)(3x – 4)
6.
c ( x ) = 2 x^3 – 3 x^2 – 14 x t(x) = (4x – 9)(x + 3) b(x) = x(x – 3)^2 ; c(x) = x(x + 2)(2x – 7) v(x) = x(x + 3)(2x – 5)
x p ( x )
-2 80
-1 21
(^2) — 5 0
0 -
—^1 2 -
(^3) — 4
1 -
1 — 15 1 4 — 5
1 — 13 0
2 24
2
(^3) — 4 66 —^15 16
x -2 -1 0 1 2 3 4 5 b ( x ) -50 -16 0 4 2 0 4 20
REFLECT
■ ■ What similarities are there among factoring polynomial functions from a table, graph, or symbolic representation? See margin.
■ ■ What differences are there among factoring polynomial functions from a table, graph, or symbolic representation? See margin.
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So for h ( x ), the linear factors can be identified from graphs, tables, or symbolic representations.
h(x) = 4x^3 – 4x^2 – 15x + 18
h(x) = (x + 2)(2x – 3)^2
DECOMPOSING POLYNOMIAL FUNCTIONS
Decomposing a polynomial function means to write it as a product of factors. Zeroes of the function can be used to identify linear factors of the polynomial function. ■ ■ In a graph, zeroes of a polynomial function can be identified from thex-coordinates of thex-intercepts. ■ ■ In a table, zeroes of a polynomial function can be identified by anx-value that generates a function value of 0. ■ ■ In a symbolic representation, some polynomial functions can be factored using algebraic methods, including: ✔ ■ Trial and Error ✔ ■ Perfect Square Trinomials ✔ ■ Difference of Squares or Cubes ✔ ■ Sum of Cubes ✔ ■ Perfect Cube Polynomials (four terms) ✔ ■ Grouping
x h ( x )
-3 -
—^1 2 18
-1 25
0 18
—^1 2 10
1 3
1 — 21 0
2 4
SYMBOLIC
x + 2 2 x – 3
x + 2
2 x – 3
GRAPH
TABLE
600 C H A P T E R 5 : A LG E B R A I C M E T H O D S
ADDITIONAL
EXAMPLES
Given the tables of values
below, write the quadratic
or cubic functions as a
product of linear factors.
Also, using the properties
of algebra, write it as a
polynomial function.
f(x) = (x + 5)(2x + 3) =
2x^2 + 13x + 15
g(x) = (x – 4)^2 = x^2 – 8x +
h(x) = (x + 3)(2x + 5)(x – 2) =
2x^3 + 7x^2 – 7x – 30
x f ( x )
-6 9 -5.5 4 -5 0 -4.5 - -4 - -3.5 - -3 - -2.5 - -2 - -1.5 0 -1 4
x g ( x )
0 16 1 9 2 4 3 1 4 0 5 1 6 4 7 9 8 16
x h ( x )
-4 - -3.5 -5. -3 0 -2.5 0 -2 - -1.5 -10. -1 - -0.5 - 0 - 0.5 -31. 1 - 1.5 - 2 0
600 C H A P T E R 5 : A LG E B R A I C M E T H O D S
EXAMPLE 1
Given the table of values, write a cubic function, f ( x ), as a product of linear factors. Graph the function to verify.
STEP 1 The table of values shows three zeroes for f ( x ). The function
values are zero when x is –2, 0, and 5. Using that information,
write the three factors of f ( x ).
x = -2 x = 0 x = 5 x + 2 = 0 x – 5 = 0
STEP 2 Write the function f ( x ) in factored form.
f ( x ) = x ( x + 2)( x – 5)
STEP 3 Graph the function. Verify that the zeroes in the graph are the
same as x -coordinates for the function values of zero in the table.
x -3 -2 -1 0 1 2 3 4 5 6 f ( x ) -24 0 6 0 -12 -24 -30 -24 0 48
YOU TRY IT! #
Given the table of values, write the quadratic function g ( x ) as a product of linear factors. Also, using the properties of algebra, write it as a polynomial function.
g(x) = (2x + 3)(2x – 3) or g(x) = 4x^2 – 9
x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 g ( x ) 7 0 -5 -8 -9 -8 -5 0 7
602 C H A P T E R 5 : A LG E B R A I C M E T H O D S
ADDITIONAL
EXAMPLES
Given the functions below,
write the quadratic or cubic
functions as a product of
linear factors.
1. n ( x ) = 27 x^3 – 125
Since n(x) is a difference of
cubes, it can be rewritten as
n(x) = (3x – 5)(9x^2 + 15x + 25).
2. p ( x ) = 16 x^2 – 72 x + 81
Since p(x) is a perfect square
trinomial, it can be rewritten as
p(x) = (4x – 9)^2.
3. q ( x ) = 64 x^3 + 336 x^2 + 588 x
Since q(x) is a perfect cube
polynomial, it can be rewritten
as q(x) = (4x + 7)^3.
602 C H A P T E R 5 : A LG E B R A I C M E T H O D S
YOU TRY IT! #
Given the graph of h ( x ), identify its zeroes and write a cubic function as a product of linear factors.
The zeroes for the function shown in the graph are –5, –1.5 or –
(^3) — 2 , and 1. Therefore, h(x) = (x + 5)(2x + 3)(x – 1).
EXAMPLE 3
Write the polynomial function q ( x ) = x^3 + 6 x^2 + 9 x as a product of linear factors.
STEP 1 Look for common factors in every term. If there are any, factor
those out first. There is a factor of x in each of the terms.
q ( x ) = x^3 + 6 x^2 + 9 x = x ( x^2 + 6 x + 9)
STEP 2 Determine if the remaining factor, ( x^2 + 6 x + 9), follows a pattern
of differences of squares or perfect square trinomials.
x^2 + 6 x + 9 follows a pattern of perfect squares trinomials: a^2 + 2 ab + b^2 = ( a + b )^2 where a is x and b is 3. So x^2 + 6 x + 9 = ( x + 3)^2.
STEP 3 Write q ( x ) = x^3 + 6 x^2 + 9 x as a product of linear factors.
q ( x ) = x ( x + 3)( x + 3) = x ( x + 3)^2.
YOU TRY IT! #4 ANSWER:
The numbers that have a product of –30 and a sum of 13 are 15 and –2. Write g(x) = 3x^2 + 13x – 10
as g(x) = 3x^2 + 15x – 2x – 10 to factor as 3x(x + 5) – 2(x + 5) and finally g(x) = (3x – 2)(x + 5).
PAIRS WITH PRODUCT OF -30^ -1, 30^ 1, -30^ -2, 15^ 2, -15^ -3, 10^ 3, - SUMS 29 -29 13 -13 7 -
ADDITIONAL EXAMPLES
Given the functions below,
write the quadratic or cubic
functions as a product of
linear factors by grouping
their terms and factoring.
1. r ( x ) = 5 x^2 + 13 x – 6
r(x) = (5x – 2)(x + 3)
2. s ( x ) = 3 x^3 + 2 x^2 – 75 x – 50
s(x) = (3x + 2)(x^2 – 25) =
(3x + 2)(x – 5)(x + 5)
3. t (x) = 4 x^3 – 24 x^2 + 9 x – 54
t(x) = (4x^2 + 9)(x – 6)
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YOU TRY IT! #
Write the polynomial function p ( x ) = 25 x^2 – 81 as a product of linear factors. This function follows the pattern of the difference of squares: p(x) = (5x – 9)(5x + 9).
EXAMPLE 4
Write the cubic function f ( x ) = 4 x^3 + 12 x^2 – x – 3 as a product of linear factors by group- ing its terms and factoring.
STEP 1 Group the terms of the cubic function and analyze them for common factors.
f ( x ) = (4 x^3 + 12 x^2 ) – ( x + 3). Notice that by grouping the last two terms, you factor -1 from both - x and -3.
STEP 2 Identify and factor out a common factor in each of the grouped terms.
f ( x ) = 4 x^2 ( x + 3) – 1( x + 3)
STEP 3 Factor out the common binomial between the two pairs of factored terms.
f ( x ) = ( x + 3)(4 x^2 – 1)
STEP 4 Determine if either factor can be written as a product of factors.
(4 x^2 – 1) is a difference of two squares. So (4 x^2 – 1) can be written as (2 x + 1)(2 x – 1).
STEP 5 Write the final factored form of the function:
f ( x ) = ( x + 3)(2 x + 1)(2 x – 1).
YOU TRY IT! #
Write the quadratic function g ( x ) = 3 x^2 + 13 x – 10 as a product of linear factors by finding those pairs of numbers that produce the product of its first and last terms’ coefficients and a sum of the middle term’s coefficient. See margin.
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g(x) = (x − 3)(x + 2)(x + 5) = x 3 + 4x^2 – 11x − 30 g(x) = (4x - 1)(2x + 1)(x + 2) = 8x^3 + 18x^2 + 3x − 2 For questions 9 – 14, write the polynomial function as a product of linear factors.
9. g ( x ) = x^2 – 9 10. h ( x ) = 4 x^2 + 16 x + 16 g(x) = (x - 3)(x + 3) h(x) = 4(x + 2) 2 11. p ( x ) = x^3 + 10 x^2 + 25 x 12. q ( x ) = 8 x^3 + 27 p(x) = x(x + 5)^2 q(x) = (2x + 3)(4x^2 – 6x + 9) 13. r ( x ) = 27 x^3 – 1 14. s ( x ) = x^3 + 4 x^2 – 4 x – 16 r(x) = (3x - 1)(9x^2 + 3x + 1) s(x) = (x - 2)(x + 2)(x + 4)
For questions 15 - 20, write the function as a product of linear factors by grouping its terms and factoring.
15. f ( x ) = 2 x^2 – 7 x + 3 16. g ( x ) = 4 x^2 + 16 x + 15 f(x) = (x − 3)(2x − 1) g(x) = (2x + 3)(2x + 5) 17. h ( x ) = 6 x^2 – 11 x − 10 18. m ( x ) = x^3 + 6 x^2 – 4 x – 24 h(x) = (2x − 5)(3x + 2) m(x) = (x − 2)(x + 2)(x + 6) 19. n ( x ) = x^3 + 8 x^2 – 9 x – 72 20. p ( x ) = x^3 + 4 x^2 – 16x – 64 n(x) = (x + 3)(x – 3)(x + 8) p(x) = (x – 4)(x + 4) 2
