- d.
(2 – 3x^3 ) – [(3x^3 + 1) – (1 – 2x^3 )]
= 2 – 3x^3 – [3x^3 + 1 – 1 + 2x^3 ]
= 2 – 3x^3 – [5x^3 ]
= 2 – 3x^3 – 5x^3
= 2 – 8x^3
- b.The degree of a polynomial is the highest
power to which the variable xis raised. For the
polynomial –5x^8 + 9x^4 – 7x^3 – x^2 , the term
involving the highest power ofxis –5x^8 , so the
degree of the polynomial is 8.
- c.For the polynomial –^32 x+ 5x^4 – 2x^2 + 12,
the term involving the highest power ofxis
5 x^4 , so the degree of the polynomial is 4.
- a.A constant polynomial is of the form cx^0 =
c,where cis a constant. By this definition, the
degree of the constant polynomial 4 is zero.
- c.By definition, a polynomial is an expression
of the form anxn+ an–1xn–1+ ... + a 1 x+ a 0
where a 0 ,a 1 , ...,anare real numbers and nis a
nonnegative integer. Put simply, once the
expression has been simplified, it cannot
contain negative powers of the variablex.
Therefore, the expression x– 3x–2is not a
polynomial.
- c.A polynomial is an expression of the form
anxn+ an–1xn–1+ ... + a 1 x+ a 0 ,where a 0 ,a 1 , ...,
anare real numbers and nis anonnegative
integer. That is, once the expression has been
simplified, it cannot contain negative powers
of the variablex. If we simplify the expression
(–2x)–1– 2 using the exponent rules, we obtain
- ^12 x–1– 2, which cannot be a polynomial
because of the term –^12 x–1. Note that the
expression given in choice aisa polynomial;
the coefficients, not the variable, involve nega-
tive exponents. The expression in choice bis a
polynomial for similar reasons; note that the
first term is really just a constant since x^0 = 1.
- d.The statements in choices a,b, and care all
true, and follow from the fact that simplifying
such arithmetic combinations of polynomials
simply involves adding and subtracting the
coefficients of like terms. Note also that, by
definition, a trinomialis a polynomial with
three terms and a binomialis a polynomial
with two terms.
- a.In general, dividing one polynomial by
another will result in an expression involving a
term in which the variable is raised to a nega-
tive power. For instance, the quotient of even
the very simple polynomials 3 and x^2 is x^32 =
3 x–2, which is not a polynomial.
- b.
–(–2x^0 )–3+ 4–2x^2 – 3–1x– 2
= –(–2)–3+ x^2 – ^13 x– 2
= – + x^2 – ^13 x– 2
= – –^1 8 + 116 x 2 – ^13 x– 2
= 116 x^2 – ^13 x– ^185
- d.
- (2 – (1 – 2x^2 – (2x^2 – 1)))– (3x^2 – (1 – 2x^2 ))
= –(2 – (1 – 2x^2 – 2x^2 + 1))– (3x^2 – 1 + 2x^2 )
= –(2 – (2 – 4x^2 ))– (5x^2 – 1)
= –(2 – 2 + 4x^2 ) – (5x^2 – 1)
= –4x^2 – 5x^2 + 1
= –94x2 + 1
^1
42
^1
(–2)^3
^1
42
ANSWERS & EXPLANATIONS–
