Classify the following as linear, quadratic and cubic polynomials:
(i) \( x^{2}+x \)
(ii) \( x-x^{3} \)
(iii) \( y+y^{2}+4 \)
(iv) \( 1+x \)
(v) \( 3 t \)
(vi) \( r^{2} \)
(vii) \( 7 x^{3} \)
To do:
We have to classify the given polynomials as linear, quadratic and cubic polynomials.
Solution:
Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.
A linear polynomial is a polynomial of degree 1.
A quadratic polynomial is a polynomial of degree 2.
A cubic polynomial is a polynomial of degree 3.
A polynomial's degree is the highest or the greatest power of a variable in a polynomial.
Therefore,
(i) In \( x^{2}+x \), the term $x^2$ has a variable of power $2$.
This implies, the degree of \( x^{2}+x \) is $2$.
Therefore, the given polynomial is a quadratic polynomial.
(ii) In \( x-x^{3} \), the term $-x^3$ has a variable of power $3$.
This implies, the degree of \( x-x^{3} \) is $3$.
Therefore, the given polynomial is a cubic polynomial.
(iii) In \( y+y^{2}+4 \), the term $y^2$ has a variable of power $2$.
This implies, the degree of \( y+y^{2}+4 \) is $2$.
Therefore, the given polynomial is a quadratic polynomial.
(iv) In \( 1+x \), the term $x$ has a variable of power $1$.
This implies, the degree of \( 1+x \) is $1$.
Therefore, the given polynomial is a linear polynomial.
(v) In \( 3t \), the term $3t$ has a variable of power $1$.
This implies, the degree of \( 3t \) is $1$.
Therefore, the given polynomial is a linear polynomial.
(vi) In \( r^{2} \), the term $r^2$ has a variable of power $2$.
This implies, the degree of \( r^{2} \) is $2$.
Therefore, the given polynomial is a quadratic polynomial.
(vii) In \( 7x^{3} \), the term $7x^3$ has a variable of power $3$.
This implies, the degree of \( 7x^{3} \) is $3$.
Therefore, the given polynomial is a cubic polynomial.
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