Degree of a Polynomial

The zero polynomial has no non-zero term, so no highest power can be identified; hence its degree is undefined. The constant polynomial \(7\) has degree 0. Exam tip: check for the zero polynomial first.

The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. Since \(c\neq0\), the term \(cx^5\) is non-zero. Thus, the highest power of \(x\) is \(5\), so the polynomial has degree \(5\). The value \(2\) is only the degree of the term \(2x^2\), not of the entire polynomial. Exam tip: Always check whether the coefficient of the highest-degree term is zero.

In \(x^4+2x\), the highest power of \(x\) is 4, so it is a polynomial of degree 4. Therefore, its degree is neither 1 nor 0. \(7x-3\) has degree 1, and \(12\) is a non-zero constant polynomial of degree 0. \(\frac{1}{x}+1\) is not a polynomial because it contains \(x^{-1}\). Exam tip: To find a polynomial's degree, identify the highest non-zero exponent of its variable.

The degree of a polynomial is the greatest exponent of the variable among terms with non-zero coefficients. In \(6y^3-y+9\), the exponents of \(y\) are 3, 1, and 0 respectively, so the greatest exponent is 3. Looking only at the term \(-y\) and choosing 1 is incorrect. Exam tip: identify the exponent in every term before selecting the greatest one.

The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. Here, \(ax^7\) has exponent 7, and its coefficient is non-zero because \(a\neq0\). Therefore, the degree is 7. The term \(bx^3\) has exponent 3, so it cannot determine the degree. Exam tip: first check that the coefficient of the highest-power term is not zero.

When \(a=0\), the term \(ax^7\) becomes zero. The polynomial is then \(bx^3+2\). Since \(b\neq0\), the coefficient of \(x^3\) is non-zero, so the highest remaining power of \(x\) is 3. Hence, its degree is 3. Option 2 may seem tempting because of the constant \(2\), but a non-zero constant has degree 0. Exam tip: Remove terms with zero coefficients before identifying the highest exponent.

\(\sqrt{5}\) is a non-zero constant polynomial because it has no term containing \(x\). Every non-zero constant polynomial has degree \(0\). The degree of \(5x\) is \(1\), while that of \(x^2+5\) is \(2\). Option A is the zero polynomial, whose degree is not defined. Exam tip: Any non-zero constant has degree \(0\).

Substituting \(b=0\) makes \(bx^5=0\), so the expression reduces to \(ax^2-4\). Since \(a\neq0\), the coefficient of \(x^2\) is non-zero. Thus, the highest power of \(x\) remaining in the polynomial is 2, so its degree is 2. Option 5 is incorrect because the \(x^5\) term becomes zero. Exam tip: first substitute the given coefficient values, remove terms with zero coefficients, and then find the highest remaining exponent.

In \(\frac{x^3}{4}-\sqrt{6}x+10\), the powers of \(x\) are \(3\), \(1\), and \(0\), all of which are non-negative integers. Therefore, it is a polynomial. Polynomial coefficients may be real numbers, so both \(\frac14\) and \(-\sqrt{6}\) are valid coefficients. The highest power of \(x\) is \(3\), so its degree is \(3\). An irrational coefficient such as \(\sqrt{6}\) does not make an expression non-polynomial. Exam tip: find the degree by identifying the greatest exponent of the variable, not by checking the type of coefficient.

In \(x^5-3x+2\), the highest power of \(x\) is \(5\), so its degree is \(5\), which is greater than \(2\). \(9x^2+1\) has degree \(2\), and \(4x-5\) has degree \(1\). \(\frac{1}{x}+2\) is not a polynomial because \(\frac{1}{x}=x^{-1}\) has a negative exponent of the variable. Exam tip: the degree of a polynomial is the highest non-negative integer exponent of its variable.

In \(3x^4-7x+6\), the highest power of \(x\) is \(4\). Therefore, it is a polynomial of degree \(4\), and since \(4<5\), option C is correct. Option A is a polynomial but has degree \(5\). In option B, \(x\) occurs in the denominator, and option D has a fractional exponent, so neither is a polynomial. Exam tip: The exponents of a variable in a polynomial must be non-negative integers.

\(5x^7\) and \(-5x^7\) are like terms and add to 0. Thus, \(p(x)=4x^3-1\). In the simplified polynomial, the greatest exponent with a non-zero coefficient is 3, so its degree is 3. Merely seeing \(x^7\) terms is not enough because they cancel. Exam tip: always simplify like terms before determining the degree.

When \(b=0\), the term \(bx^6\) becomes zero, so the polynomial is \(ax^8-5x+2\). Since \(a\neq0\), the coefficient of \(x^8\) is non-zero. The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient; therefore, the degree is 8. Option 6 is incorrect because the \(x^6\) term vanishes. Exam tip: First remove all terms whose coefficients are zero.

On substituting \(a=0\), \(ax^8=0\), so the \(x^8\) term disappears. Since \(b\neq0\), \(bx^6\) remains a non-zero term. The remaining terms have powers \(1\) and \(0\), so the highest power is \(6\). Option 8 is incorrect because its coefficient becomes zero. Exam tip: Remove terms with zero coefficients before finding the degree.

The expression is \(\frac{1}{3}x^4-\sqrt{10}x^2+7\). In a polynomial, the powers of the variable must be non-negative integers. Its coefficients may be fractional or irrational real numbers. The highest power of \(x\) is \(4\), so it is a polynomial of degree \(4\). Both \(\frac{1}{3}\) and \(-\sqrt{10}\) are coefficients, so options A and D are incorrect. Exam tip: find the degree from the highest exponent of the variable, not from the type of coefficient.

In \(x^6-2x^2+9\), the highest power of \(x\) is \(6\), so its degree is \(6\), which is greater than \(3\). \(2x^3+1\) has degree \(3\), and \(5x-4\) has degree \(1\). \(\frac{1}{x}+6=x^{-1}+6\) is not a polynomial because it contains a negative power of \(x\). Exam tip: Find a polynomial's degree by identifying the greatest non-negative integer power of its variable.

For degree (5), the coefficient of (x^8) must be (0) and the coefficient of (x^5) must be non-zero. This condition holds at (a=2).

First expand: \(x^3(x^4-2)=x^7-2x^3\). Therefore, the expression becomes \(x^7-2x^3-x^7+5x^2=-2x^3+5x^2\). The highest power of \(x\) in the simplified polynomial is 3, so its degree is 3. Option 7 is incorrect because \(x^7\) and \(-x^7\) cancel. Exam tip: Always simplify and combine like terms before finding the degree.

On substituting \(k=3\), we get \(k^2-9=9-9=0\) and \(k-3=0\). Thus, the \(x^6\) and \(x^4\) terms vanish, leaving \(q(x)=2x^2\). Its highest power of \(x\) is 2, so the degree is 2. Option 6 may seem tempting from the original expression, but its coefficient becomes zero when \(k=3\). Exam tip: substitute the parameter value and remove zero-coefficient terms before finding the degree.

The zero polynomial has no non-zero term. Degree is determined by the greatest exponent among the non-zero terms, and there is no such term here; therefore, its degree is not defined. Option A applies to a non-zero constant polynomial such as 5, not to the zero polynomial. Exam tip: always distinguish the zero polynomial from a non-zero constant polynomial.

Combining the like terms gives \(x^5-3x^5+2x^5=(1-3+2)x^5=0\). Hence, \(f(x)=4x-1\). The highest power of \(x\) in the simplified polynomial is 1, so its degree is 1. Option 5 may seem tempting if the expression is not simplified, but all the \(x^5\) terms cancel out. Exam tip: always simplify by combining like terms before finding the degree of a polynomial.

In a polynomial of several variables, the total degree of a term is the sum of the exponents of all its variables. Here, the total degrees of the terms are 2+5=7, 4+2=6, 1+1=2, and 0 respectively. The greatest total degree is 7, so the polynomial has total degree 7. Although 6 is the degree of the second term, it is not the highest. Exam tip: Add the exponents in each term and choose the greatest sum.

To find the degree with respect to x, treat y as part of the coefficient. The powers of x in the given expression are 3, 5, 0, and 0. The greatest power is 5, so the answer is 5. The number 7 is the power of y in 6x^3y^7, not the power of x. Exam tip: compare powers only of the variable with respect to which the degree is asked.

The degree of a polynomial with respect to y is the greatest exponent of y. In the given expression, the powers of y in the terms are 7, 2, 1, and 0. The greatest power is 7, so the answer is 7. The powers of x are treated as part of the coefficients when considering y; hence 5 is not the degree. Exam tip: When degree is asked with respect to one variable, compare powers of that variable only.

The governing concept is that the degree of a polynomial is the greatest exponent of x with a non-zero coefficient. In 2x^n + 5x^7 - 3, the coefficient of x^n is 2, which is non-zero. The question additionally states n > 7, so the x^n term has a higher power than x^7 and must determine the degree. We therefore need an option whose value is greater than 7. Among the choices, only n = 8 satisfies that condition, and then the polynomial has degree 8. Options n = 5 and n = 3 violate n > 7, while n = 7 does not satisfy the strict inequality and would also make the highest powers equal rather than establish the stated condition. Hence option C is the unique valid answer.