Polynomial Vocabulary: Problems with Solutions

Solution:
The terms contain different powers of [tex]x[/tex], so there are no like terms. Write the polynomial in standard form (descending powers of [tex]x[/tex]):
[tex]5x^2-3x^5+2x-5=-3x^5+5x^2+2x-5[/tex]
The degree of a polynomial is the highest degree of its terms. The highest power of [tex]x[/tex] is in the term [tex]-3x^5[/tex], so the degree of the polynomial is [tex]5[/tex].

Solution:
The powers of [tex]y[/tex] in the three summands are different, so there are no like terms and the polynomial is already in standard form. Its terms are:
[tex]2y^6, \quad -\frac{3}{2}y^2, \quad 1[/tex]
So the polynomial has [tex]3[/tex] terms (it is a trinomial); the number [tex]1[/tex] is its constant term.

Solution:
The numbers [tex]\frac{5}{2}[/tex] and [tex]\frac{1}{2}[/tex] are like terms. Combine them and write the polynomial in standard form:
[tex]\frac{5}{2}-x+5x^4-3x^2+\frac{1}{2}=5x^4-3x^2-x+3[/tex]
The highest power of [tex]x[/tex] is in the term [tex]5x^4[/tex], so the degree of the polynomial is [tex]4[/tex].

Solution:
All powers of [tex]x[/tex] are different, so there are no like terms. In standard form:
[tex]4x^3-4x^5+4x+4=-4x^5+4x^3+4x+4[/tex]
The polynomial has [tex]4[/tex] terms: [tex]-4x^5[/tex], [tex]4x^3[/tex], [tex]4x[/tex] and [tex]4[/tex]. The highest power of [tex]x[/tex] is [tex]5[/tex] (in the term [tex]-4x^5[/tex]), so the degree is [tex]5[/tex], not [tex]3[/tex]: the degree is not given by the first written term, but by the highest power among all terms.

Solution:
The name shows the number of terms: a monomial has one term, a binomial two terms and a trinomial three terms. So a polynomial with exactly three terms is a trinomial, for example [tex]x^2-5x+6[/tex].
The degree is not related to this name: [tex]x^2-5x+6[/tex] is a trinomial of degree [tex]2[/tex], while [tex]3x^3+3[/tex] has degree [tex]3[/tex], leading coefficient [tex]3[/tex] and constant term [tex]3[/tex], but it is a binomial, not a trinomial.

Solution:
Compute the bracket and write the polynomial in standard form:
[tex]5x^2-x^4+(3-5)=-x^4+5x^2-2[/tex]
The constant term is the term that does not contain [tex]x[/tex], so it is [tex]-2[/tex]. The polynomial is a trinomial of degree [tex]4[/tex] with leading coefficient [tex]-1[/tex].

Solution:
A polynomial of degree [tex]1[/tex] in [tex]y[/tex] has the form [tex]ay+b[/tex], where [tex]a\ne 0[/tex] is the leading coefficient and [tex]b[/tex] is the constant term. Here [tex]a=1[/tex] and [tex]b=0[/tex]:
[tex]1\times y+0=y[/tex]
The other options do not fit: [tex]y+1[/tex] has constant term [tex]1[/tex], [tex]y^2[/tex] has degree [tex]2[/tex], and the number [tex]1[/tex] has degree [tex]0[/tex].

Solution:
A binomial is a polynomial with exactly [tex]2[/tex] terms (the prefix "bi" means "two"; a monomial has [tex]1[/tex] term and a trinomial [tex]3[/tex] terms).
For example, [tex]3x^2-5[/tex] is a binomial.

Solution:
A polynomial is a sum of monomials: each term is a number (coefficient) multiplied by a power of the variable whose exponent is a non-negative integer. The coefficients may be any real numbers, including irrational ones.
In [tex]\sqrt{2}x^3-\sqrt[3]{7}x+\frac{3}{2}[/tex] the coefficients [tex]\sqrt{2}[/tex], [tex]-\sqrt[3]{7}[/tex] and [tex]\frac{3}{2}[/tex] are real numbers and the exponents of [tex]x[/tex] are [tex]3[/tex], [tex]1[/tex] and [tex]0[/tex], so it is a polynomial of degree [tex]3[/tex].
The other expressions are not polynomials: in [tex]\sqrt{x}[/tex] the variable is under a root, in [tex]\frac{2}{x^2-5x+3}[/tex] the variable is in the denominator, and [tex]-2x^{-3}=-\frac{2}{x^3}[/tex] has a negative exponent.

Solution:
The coefficients of a polynomial can be any real numbers, including irrational ones such as [tex]\pi[/tex], [tex]e[/tex], [tex]\sqrt{5}[/tex] or [tex]\sqrt{2}-\sqrt{3}[/tex]. Only the variable is restricted: its exponents must be non-negative integers, and it must not be under a root or in a denominator.
[tex]A[/tex] is a polynomial of degree [tex]3[/tex] in [tex]x[/tex] with constant term [tex]-\pi[/tex]; [tex]B[/tex] is a polynomial of degree [tex]5[/tex] in [tex]x[/tex] (the root is taken of the number [tex]5[/tex], not of [tex]x[/tex]); [tex]C[/tex] is a polynomial of degree [tex]4[/tex] in [tex]y[/tex] with coefficients [tex]\pi[/tex], [tex]-e[/tex], [tex]\sqrt{2}-\sqrt{3}[/tex] and [tex]1[/tex].
So all three expressions are polynomials.

Solution:
Expand using the formula [tex](u+v)^2=u^2+2uv+v^2[/tex]:
[tex](2x+3yz)^2=4x^2+12xyz+9y^2z^2[/tex]
The degree of a monomial is the sum of the exponents of all its variables: [tex]4x^2[/tex] has degree [tex]2[/tex], [tex]12xyz[/tex] has degree [tex]1+1+1=3[/tex] and [tex]9y^2z^2[/tex] has degree [tex]2+2=4[/tex]. There are no like terms, so the degree of the polynomial is the largest of these degrees, namely [tex]4[/tex].

Solution:
Combine the like terms [tex]ax^5[/tex] and [tex]x^5[/tex] to get the standard form
[tex]P(x)=(a+1)x^5-x^4-3ax^2+ax-a[/tex]
The first-degree term is [tex]ax[/tex], so [tex]a=6[/tex]. Then the coefficient of [tex]x^5[/tex] is [tex]a+1=7\ne 0[/tex] and the polynomial has degree [tex]5[/tex]:
[tex]P(x)=7x^5-x^4-18x^2+6x-6[/tex]
Its leading coefficient is [tex]7[/tex].

Solution:
Two polynomials are identically equal exactly when the coefficients of like powers of [tex]x[/tex] are equal. The coefficients of [tex]x^2[/tex] (both are [tex]-3[/tex]) and the constant terms (both are [tex]-5[/tex]) already agree. Equate the coefficients of [tex]x^3[/tex] and of [tex]x[/tex]:
[tex]a+1=2\Rightarrow a=1[/tex]
[tex]2b+3=7\Rightarrow b=2[/tex]

Solution:
When polynomials are multiplied, the highest-degree term of the product is the product of the highest-degree terms of the factors (all other products of terms have a smaller degree, so nothing can cancel it). The highest-degree term of [tex]x^3-3x^2[/tex] is [tex]x^3[/tex] and that of [tex]2x-3[/tex] is [tex]2x[/tex]. Hence the highest-degree term of [tex]P(x)[/tex] is
[tex](x^3)^4\times (2x)^5=x^{12}\times 32x^5=32x^{17}[/tex]
So [tex]P(x)[/tex] has degree [tex]17[/tex] and its leading coefficient is [tex]32[/tex].

Solution:
Two polynomials are equal when their standard forms coincide, i.e. when the coefficients of like powers of [tex]x[/tex] are equal. Bring both polynomials to standard form:
[tex]P(x)=(4x^2-4x+1)-(x^2+4x+4)+3x-1=3x^2-5x-4[/tex]
[tex]Q(x)=(3x^2+2x-3x-2)-(4x+2)=3x^2-5x-4[/tex]
In both polynomials the coefficient of [tex]x^2[/tex] is [tex]3[/tex], the coefficient of [tex]x[/tex] is [tex]-5[/tex] and the constant term is [tex]-4[/tex]; hence [tex]P(x)=Q(x)[/tex].