Section 5.1 Solutions
1) 𝑓(𝑥) = (𝑥 − 3)2 (𝑥 + 1)
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
(x-3)2(x+1) = 0
(x-3)(x-3)(x+1) = 0
x–3=0
or x – 3 = 0
x+1=0
x=3
or x = 3
or x -1
Answer: zeros are x = 3 multiplicity even x = -1 multiplicity odd
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: touches the x-axis at x = 3, crosses the x-axis at x = -1
c) Determine the maximum number of turning points on the graph
if you multiply the firsts in each parenthesis you get x*x*x = x3, hence this is a third degree polynomial
Answer: has at most 2 turning points
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer: Will behave like g(x) = x3
e) Describe the end behavior
Answer: falls to the left, rises to the right
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
Answer: increasing (−∞, . 𝟑𝟑) ∪ (𝟑, ∞) decreasing (. 𝟑𝟑, 𝟑)
3) 𝑓(𝑥) = (2𝑥 − 3)2 (𝑥 − 1)1
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
(2x-3)(2x-3)(x-1) = 0
Enough to solve 2x – 3 = 0 and x – 1 = 0
2x – 3 = 0
x -1=0
x = 3/2
x=1
Answer: zeros are x = 3/2 has even multiplicity and x = 1 odd multiplicity
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: graph touches x-axis at x = 3/2 and crosses x-axis at x = 1
c) Determine the maximum number of turning points on the graph
Multiply the firsts in each parenthesis of the expanded polynomial
(2x)(2x)(x) = 4x3, this is a 3rd degree polynomial
Answer: has at most 2 turning points
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer: g(x) = 4x3
e) Describe the end behavior
Answer: falls to the left, rises to the right
f) sketch a graph and approximate the turning points, also label the x-intercepts
window used
x-min -3
x-max 3
ymin -1
ymax 1
g) state the intervals where the function is increasing and decreasing
Answer: increasing (−∞, 𝟏. 𝟏𝟕) ∪ (𝟏. 𝟓, ∞) decreasing (1.17, 1.5)
5) 𝑓(𝑥) = (𝑥 2 − 3)2 (2𝑥 + 1)
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
(x2 – 3)(x2 – 3)(2x+1) = 0
𝑥2 − 3 − 0
𝑥2 = 3
2𝑥 + 1 = 0
2𝑥 = −1
𝑥 = ±√3
𝑥=
−1
2
Answer: 𝒙 = √𝟑 𝒉𝒂𝒔 𝒆𝒗𝒆𝒏 𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒊𝒕𝒚, 𝒙 = −√𝟑 𝒉𝒂𝒔 𝒆𝒗𝒆𝒏 𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒊𝒕𝒚, 𝒙 =
−𝟏
𝟐
𝒉𝒂𝒔 𝒐𝒅𝒅 𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒊𝒕𝒚
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: graph touches x-axis at √𝟑 𝒂𝒏𝒅 − √𝟑 𝒄𝒓𝒐𝒔𝒔𝒆𝒔 𝒙 − 𝒂𝒙𝒊𝒔 𝒂𝒕 𝒙 =
−𝟏
𝟐
c) Determine the maximum number of turning points on the graph
Multiply the firsts in the expanded polynomial (x2)(x2)(2x) =2x5 this is a 5th degree polynomial
Answer: will have at most 4 turning points
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer: will behave like g(x) = 2x5
e) Describe the end behavior
Answer: Falls to the left rises to the right
Window
x-min -10
x-max 10
y-min – 10
y-max 20
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
Answer: increasing: (−∞, −√𝟑) ∪ (−𝟏, . 𝟔) ∪ (√𝟑, ∞) decreasing (−√𝟑, −𝟏) ∪ (. 𝟔, √𝟑)
7) 𝑓(𝑥) = (𝑥 2 − 6)2 (2𝑥 − 12)2
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
(x2 – 6)(x2 -6)(2x-12)(2x-12) = 0
x2 – 6 = 0
x2 = 6
𝑥 = ±√6
2x – 12 = 0
2x = 12
x=6
Answer: 𝒙 = √𝟔 𝒉𝒂𝒔 𝒆𝒗𝒆𝒏 𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒊𝒕𝒚, 𝒙 = −√𝟔 𝒉𝒂𝒔 𝒆𝒗𝒆𝒏 𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒊𝒕𝒚, 𝒙 =
𝟔 𝒉𝒂𝒔 𝒆𝒗𝒆𝒏 𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒊𝒕𝒚
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: Graph touches x-axis at all x-intercepts
c) Determine the maximum number of turning points on the graph
Multiply the firsts in the expanded polynomial (x2)(x2)(2x)(2x) = 4x6, this is a 6th degree polynomial
Answer: has at most 5 turning points
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer: will behave like g(x) = 4x6
e) Describe the end behavior
Answer: rises to the left and rises to the right
Window used
X min -10
X max 10
Y min -1000
Y max 10000
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
answer: 𝒊𝒏𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈 (−√𝟔, −. 𝟒𝟓) ∪ (√𝟔, 𝟒. 𝟒𝟓) ∪ (𝟔, ∞)𝒅𝒆𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈 (−∞, −𝟔) ∪ (−. 𝟒𝟓, √𝟔) ∪
(𝟒. 𝟒𝟓, 𝟔)
9) 𝑓(𝑥) = (𝑥 2 + 3) (2𝑥 + 1)2
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
(x2 + 3)(2x+1)(2x+1) = 0
x2 + 3 = 0
2x + 1 = 0
x2 = -3
2x = -1
𝑥 = ±√−3
x = -1/2
(this gives an i, thus doesn’t give an x-intercept)
Answer: x = -1/2, even multiplicity
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: graph touches x-axis at x = -1/2
c) Determine the maximum number of turning points on the graph
Multiply the firsts (x2)(2x)(2x) = 4x4, this is a 4th degree polynomial
Answer: has at most 3 turning points
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer: will behave like g(x) = 4x4
e) Describe the end behavior
Answer: rises to the left and rises to the right
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
𝟏
𝟏
answer: increasing (− 𝟐 , ∞) decreasing (−∞, − 𝟐)
11) 𝑓(𝑥) = (𝑥 2 + 6)(2𝑥 − 12)2
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
(x2+6)(2x-12)(2x-12) = 0
x2 + 6 = 0
x2 = -6
2x – 12 = 0
2x = 12
𝑥 = ±√−6
x=6
this reduces to something with an i and does not give an intercept
Answer: x = 6, even multiplicity
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: Graph touches the x-axis at x = 6
c) Determine the maximum number of turning points on the graph
multiply the firsts in the expanded polynomial (x2)(2x)(2x) = 4x4, this is a 4th degree polynomial
Answer: has at most 3 turning points
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer: will behave like g(x) = 4x4
e) Describe the end behavior
Answer: rises to the left, rises to the right
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
answer: increasing (𝟔, ∞) decreasing (−∞, 𝟔)
13) 𝑓(𝑥) = 𝑥 2 + 6𝑥 − 7
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
x2 + 6x – 7 = 0
(x+7)(x-1) = 0
x+ 7 = 0
x–1=0
answer x = -7, odd multiplicity
x = 1, odd multiplicity
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: graph crosses x-axis at x =-7 and at x = 1
c) Determine the maximum number of turning points on the graph
this is a 2nd degree polynomial
Answer has at most 1 turning point.
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer: like g(x) = x2
e) Describe the end behavior
Answer: rise to left and rise to the right
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
increasing (−𝟑, ∞) decreasing (−∞, −𝟑)
15) 𝑓(𝑥) = 𝑥 2 − 4
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
x2 – 4 = 0
(x+2)(x-2) = 0
x+2 = 0
x-2 = 0
Answer: x = -2 odd multiplicity
x = 2 odd multiplicity
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: graph crosses x-axis at both x = 2 and x = -2
c) Determine the maximum number of turning points on the graph
this is a second degree polynomial
Answer will have at most 1 turning point.
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer: will behave like g(x) = x2
e) Describe the end behavior
Answer: rises to the left and rises to the right
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
increasing (𝟎, ∞) decreasing (−∞, 𝟎)
17) 𝑓(𝑥) = −5𝑥 3 − 13𝑥 2 − 6𝑥
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
-5x3 – 13x2 – 6x = 0
-x(5x2 +13x + 6) = 0
-x(5x + 3)(x + 2) = 0
-x = 0
5x + 3 = 0
x+2 = 0
Answer: x = 0 x = -3/5
x = -2 (all zeros have odd multiplicity)
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: graph will cross x-axis at each 0.
c) Determine the maximum number of turning points on the graph
This is a 3rd degree polynomial
Answer has at most 2 turning points.
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer will behave like g(x) = -5x3 for large values of x.
e) Describe the end behavior
Answer: rise to the left fall to the right
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
decreasing (−∞, −𝟏. 𝟒𝟓) ∪ (−. 𝟐𝟕, ∞) increasing (-1.45, -.27)
19) 𝑓(𝑥) = −3𝑥 4 + 12𝑥 2
a) List each x-intercept and (zero) its multiplicity (round to 2 decimal places when needed)
-3x4 + 12x2 = 0
-3x2(x2 – 4) = 0
-3x2 = 0 x2 – 4 = 0
(divide by -3)
x2 = 0
xx = 0
(x+2)(x-2) = 0
x+2=0
x- 2 = 0
Answer x = 0 has even multiplicity, x = -2 and x = 2 have odd
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: graph touches at 0 and crosses at -2 and crosses at 2
c) Determine the maximum number of turning points on the graph
This is a 4th degree polynomial
answer: has at most 3 turning points
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer: graph will behave like g(x) = -3x4
e) Describe the end behavior
Answer: falls to the left, falls to the right
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
increasing (−∞, −𝟏. 𝟒𝟏) ∪ (𝟎, 𝟏. 𝟒𝟏) decreasing (-1.41, 0) (𝟏. 𝟒𝟏, ∞)
21) 𝑓(𝑥) = 5𝑥 4 + 13𝑥 3 + 6𝑥 2
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
5𝑥 4 + 13𝑥 3 + 6𝑥 2
x2(5x2 + 13x + 6) = 0
x2(5x+3)(x+2) = 0
=0
xx(5x+3)(x+2) = 0
x=0
5x+3 = 0
x+2 = 0
answer: x = 0 has even multiplicity, x = -3/5 has odd and x = - 2 has odd
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: graph touches x-axis at x = 0 and crosses x-axis at x = -3/5 and crosses at x = -2
c) Determine the maximum number of turning points on the graph
This is a 4th degree polynomial
Answer: it has at most 3 turning points
d) Which function will the graph of f(x) behave like for large values of |𝑥|
Answer: graph will behave like g(x) = 5x4
e) Describe the end behavior
Answer: rise to left and rise to the right
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
increasing (-1.57, - .38) ∪ (𝟎, ∞) decreasing (−∞, −𝟏. 𝟓𝟕) ∪ (−. 𝟑𝟖, 𝟎)
#23 – 32: Form a polynomial whose x-intercepts (zero) are given. Answers can vary depending on
choice of a leading coefficient. (multiply out your polynomial)
23) x-intercepts: 2, 3, -4; degree 3
Set x equal to each intercept
x= 2
x=3
x = -4
Move over to get
x–2=0
x–3=0
x+ 4 = 0
the desired polynomial can be found by multiplying these together
f(x) = (x-2)(x-3)(x+4)
FOIL The first parenthesis
f(x) = (x2 – 5x + 6)(x+4)
f(x) = x3 + 4x2 – 5x2 – 20x + 6x + 24
Answer: f(x) = x3 – x2 – 14x + 24
1
25) x-intercepts: 2, 3, 2; degree 3
Set x equal to each intercept
x=2
x =3
1
x =2
(multiply by 2 to clear fraction)
2x = 1
Set to 0
x–2=0
x–3=0
Multiply factors to get answer
f(x) = (x-2)(x-3)(2x-1)
f(x) = (x2 – 5x + 6)(2x – 1)
f(x) = 2x3 – 1x2 – 10x2 + 5x + 12x – 6
Answer: f(x) = 2x3 -11x2 + 17x - 6
2x – 1 = 0
27) x-intercepts:
−1
,
3
3 ; degree 2
Set x equal to each intercept, then set to zero
𝑥=
−1
3
𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑦 3 𝑡𝑜 𝑐𝑙𝑒𝑎𝑟 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛
3x = -1
x=3
3x +1 = 0
x–3=0
x=3
f(x) = (3x+1)(x-3)
Answer: f(x) = 3x2 – 8x - 3
29) x-intercepts ±√2; degree 2
Set x equal to each intercept, then set to zero
𝑥 = √2
𝑥 = −√2
𝑥 − √2 = 0
𝑥 + √2 = 0
Multiply
f(x) = (x+√2)(x-√2) the outers and inners will canel
f(x) = x2- √4
Answer: f(x) = x2 - 2
31) x-intercepts ±√2 , 5; degree 3
Set x equal to each intercept
𝑥 = √2
𝑥 = −√2
x=5
𝑥 + √2 = 0
x–5=0
Set to zero
𝑥 − √2 = 0
Multiplyf(x) = (𝑥 − √2)(𝑥 + √2)(𝑥 − 5) see problem 29 for work on multiplying first parenthesis
f(x) = (x2 – 2)(x-5)
f(x) = x3 – 5x2 - 2x + 10
Answer: f(x) = x3 – 5x2 – 2x + 10