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Write each of the following polynomials in the standard form. Also, write their degree.
(i) $x^2+3+6x+5x^4$
(ii) $a^2+4+5a^6$
(iii) $(x^3-1)(x^3-4)$
(iv) $(a^3-\frac{3}{8})(a^3+\frac{16}{17})$
(v) $(a+\frac{3}{4})(a+\frac{4}{3})$
Given:
The given expressions are:
(i) $x^2+3+6x+5x^4$
(ii) $a^2+4+5a^6$
(iii) $(x^3-1)(x^3-4)$
(iv) $(a^3-\frac{3}{8})(a^3+\frac{16}{17})$
(v) $(a+\frac{3}{4})(a+\frac{4}{3})$
To do:
We have to write the standard form of the given polynomials and find their degree.
Solution:
Polynomials:
Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.
The standard form of the polynomial is a polynomial where the terms are arranged in descending order by degree.
Degree of a polynomial:
The degree of a polynomial is the highest or the greatest power of a variable in the polynomial expression.
To find the degree, identify the exponents on the variables in each term, and add them together to find the degree of each term.
(i) The given polynomial is $x^2+3+6x+5x^4$.
The standard form of the given polynomial is $5x^4+x^2+6x+3$
The power of $x$ in $5x^4$ is $4$.
Therefore,
The degree of the given polynomial is $4$.
(ii) The given polynomial is $a^2+4+5a^6$.
The standard form of the given polynomial is $5a^6+a^2+4$
The power of $a$ in $5a^6$ is $6$.
Therefore,
The degree of the given polynomial is $6$.
(iii) The given polynomial is $(x^3-1)(x^3-4)$.
$(x^3-1)(x^3-4)=x^3(x^3-4)-1(x^3-4)$
$(x^3-1)(x^3-4)=x^6-4x^3-x^3+4$
$(x^3-1)(x^3-4)=x^6-5x^3+4$
The standard form of the given polynomial is $x^6-5x^3+4$
The power of $x$ in $x^6$ is $6$.
Therefore,
The degree of the given polynomial is $6$.
(iv) The given polynomial is $(a^3-\frac{3}{8})(a^3+\frac{16}{17})$.
$(a^3-\frac{3}{8})(a^3+\frac{16}{17})=a^3(a^3+\frac{16}{17})-\frac{3}{8}(a^3+\frac{16}{17})$
$(a^3-\frac{3}{8})(a^3+\frac{16}{17})=a^6+\frac{16}{17}a^3-\frac{3}{8}a^3-(\frac{3}{8})(\frac{16}{17})$
$(a^3-\frac{3}{8})(a^3+\frac{16}{17})=a^6+\frac{16\times8-17\times3}{17\times8}a^3-\frac{3\times16}{8\times17}$
$(a^3-\frac{3}{8})(a^3+\frac{16}{17})=a^6+\frac{128-51}{136}a^3-\frac{3\times2}{1\times17}$
$(a^3-\frac{3}{8})(a^3+\frac{16}{17})=a^6+\frac{77}{136}a^3-\frac{6}{17}$
The standard form of the given polynomial is $a^6+\frac{77}{136}a^3-\frac{6}{17}$
The power of $a$ in $a^6$ is $6$.
Therefore,
The degree of the given polynomial is $6$.
(v) The given polynomial is $(a+\frac{3}{4})(a+\frac{4}{3})$.
$(a+\frac{3}{4})(a+\frac{4}{3})=a(a+\frac{4}{3})+\frac{3}{4}(a+\frac{4}{3})$
$(a+\frac{3}{4})(a+\frac{4}{3})=a^2+\frac{4}{3}a+\frac{3}{4}a+(\frac{4}{3})\times(\frac{3}{4})$
$(a+\frac{3}{4})(a+\frac{4}{3})=a^2+\frac{4\times4+3\times3}{12}a+1$
$(a+\frac{3}{4})(a+\frac{4}{3})=a^2+\frac{16+9}{12}a+1$
$(a+\frac{3}{4})(a+\frac{4}{3})=a^2+\frac{25}{12}a+1$
The standard form of the given polynomial is $a^2+\frac{25}{12}a+1$
The power of $a$ in $a^2$ is $2$.
Therefore,
The degree of the given polynomial is $2$.
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