Degree of a Polynomial

In \(5x^4-x+6\), the highest power of \(x\) is \(4\). Therefore, it is a fourth-degree polynomial, not a cubic polynomial. \(2x^3+x+1\) is cubic because its highest power is \(3\). In \(\sqrt{x}+x^3\), \(\sqrt{x}=x^{1/2}\), while \(x^{-3}+x^2\) contains a negative exponent, so neither is a polynomial. Exam tip: the exponents of a variable in a polynomial must be \(0\) or positive integers.

The numerator has a common factor \(x\): \(x^3+2x=x(x^2+2)\). Therefore, for \(x\ne0\), \(\frac{x^3+2x}{x}=x^2+2\). The highest power of \(x\) is 2, so it is a quadratic polynomial. A linear polynomial has highest power 1. Exam tip: simplify the expression first, then use the highest exponent of the variable to identify the degree.

\(x^5+x^3-x+2\) has four terms: \(x^5\), \(x^3\), \(-x\), and \(2\). The highest power of \(x\) is \(5\), so the degree of this polynomial is \(5\). Option B has four terms but degree \(4\). Option C has degree \(5\), but it contains only two terms. Option D is not a polynomial because \(\frac{1}{x}=x^{-1}\) has a negative exponent. Exam tip: First count the terms, then identify the greatest non-negative integer exponent of the variable.

The degree of a polynomial is the greatest power of the variable with a non-zero coefficient. Here, the coefficient of \(x^4\) is \(a\), and \(a\neq0\), so the \(x^4\) term remains present. No value of \(b\) or \(c\) can affect this highest-power term; therefore, the polynomial has degree 4. It could be quadratic only if \(a=0\). Exam tip: Before identifying a polynomial’s degree, always check the coefficient of its highest-power term.

The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. Since \(a\neq0\), the term \(ax^5\) is present, so the highest power of \(x\) is \(5\). The term \(bx^3\) has exponent \(3\), so it does not determine the degree. Exam tip: always verify that the coefficient of the highest-power term is non-zero.

The degree of a polynomial is the greatest exponent of the variable having a non-zero coefficient. Since \(a\neq0\), the term \(ax^7\) is present. Hence, the degree of \(p(x)\) is \(7\). It is not \(2\), because \(7\) is the greater exponent. Exam tip: always verify that the coefficient of the highest-power term is non-zero.

In \(x^3+1\), the powers of \(x\) are 3 and 0, both non-negative integers, so it is a polynomial in \(x\). Its highest power is 3, making it a cubic polynomial rather than a linear polynomial. \(5x-2\) is linear because its degree is 1. In \(\frac{1}{x}+2\) and \(\sqrt{x}+1\), the powers of \(x\) are \(-1\) and \(\frac12\), so they are not polynomials. Exam tip: variable exponents in a polynomial must be 0 or positive integers.

For degree (3), both (x^{11}) and (x^7) terms must vanish. At (a=2), both higher-term coefficients become (0).

The expression must first be expanded and like terms must then be combined. Multiplying \(x^4\) by each term inside the bracket gives \(x^4\cdot x^5=x^9\) and \(x^4(-3x)=-3x^5\). Therefore the full expression becomes \(x^9-3x^5-x^9+6x^6-2\). The two terms with power 9 have opposite coefficients and cancel completely.

After cancellation, the expression is \(6x^6-3x^5-2\). The highest remaining exponent is 6, and its coefficient 6 is nonzero, so the degree is 6. Hence option B is correct. Choosing 9 would ignore the cancellation; this is why simplifying before deciding the degree is essential.

On substituting k=1, we get (k-1)^2=0 and k^2-1=0. Hence the coefficients of the x^{10} and x^6 terms become zero, leaving q(x)=4x^2. The degree of a polynomial is the highest exponent with a non-zero coefficient, so its degree is 2. Option 0 would apply if the polynomial reduced to a non-zero constant. Exam tip: substitute the parameter first, discard zero-coefficient terms, and then identify the highest remaining exponent.

The degree is the greatest exponent of the variable in any term, not the position of a term. Here the exponents are 0, 3, 6, and 8, so the degree is 8. The middle term \(x^6\) does not decide it. Exam tip: list exponents and select the largest one.

Combining like terms gives \(4x^8-9x^8+5x^8=(4-9+5)x^8=0\). Thus, \(f(x)=-3x^4+10\). The highest exponent with a non-zero coefficient is \(4\), so the degree of the polynomial is \(4\). Option 8 is incorrect because all the \(x^8\) terms cancel out. Exam tip: simplify like terms before finding a polynomial's degree.

The total degree of a multivariable polynomial is the greatest sum of the exponents of the variables in any one term. Here, the term degrees are 5+4=9, 3+7=10, 1+2=3, and 0 for the constant term -11. Therefore, the highest total degree is 10. Option 9 is the degree of the first term, not of the whole polynomial. Exam tip: Add the exponents in each term and select the largest 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 8, 6, 0, and 0. The greatest of these is 8, so the degree of the polynomial in x is 8. The number 9 is a power of y, not of x. Exam tip: When the degree is asked with respect to one variable, consider powers of that variable only.

In \(4x^3y^2\), the sum of exponents is \(3+2=5\), so its total degree is 5. However, option A also has \(3x^4y\) of total degree 5, so it would also be correct. In exams, check exponent sums for every term.

The degree of a polynomial is the highest power of the variable with a non-zero coefficient. Since the term \(3x^9\) is present and \(m<9\), the power in \(6x^m\) must be less than 9. For \(m=7\), the highest power is still \(9\), so the degree remains \(9\). Although \(m=9\) would also give degree 9, it does not satisfy \(m<9\). Exam tip: While finding degree, always check any condition given on the exponent as well.

On removing brackets, \(P(x)=x^4-3x^2+1-x^4-2x^2+5=-5x^2+6\). The \(x^4\) terms cancel, so the highest remaining power is 2. Exam tip: simplify fully before finding degree.

A non-zero constant polynomial such as 7 = 7x⁰ has highest exponent 0, so its degree is 0. The degree of the zero polynomial is not defined. Exam tip: in a sum, leading terms can cancel, reducing the degree.

The degree is determined by the greatest exponent whose coefficient is nonzero. Here the highest-power term is \\( (b+4)x^{14}\\). Because \\(b\\ne-4\\), the coefficient \\(b+4\\) is not zero, so the term remains in the polynomial. Consequently, the greatest effective exponent is 14, and option A is correct.

The remaining terms are \\(8x^5\\) and \\(-1\\), with degrees 5 and 0. Neither can affect the fact that a nonzero degree-14 term is present. If \\(b=-4\\), the degree-14 term would vanish and the degree would become 5, but that case is specifically excluded. Therefore the degree is 14.

On substituting \(b=-4\), we get \(b+4=0\). Hence, the term \((b+4)x^{14}\) becomes zero, and the expression reduces to \(8x^5-1\). The highest power of \(x\) in the remaining polynomial is \(5\), so its degree is \(5\). Although \(14\) appears in the original expression, that term is not counted because its coefficient becomes zero. Exam tip: check whether the coefficient of the highest-power term becomes zero after substitution.

In a product, the leading terms multiply, so their exponents add: \(\deg(pq)=\deg p+\deg q\). In a sum, leading terms can cancel; remember this distinction in exams.

On simplifying, \(5x^7(x^2-1)-5x^9+2x^6=5x^9-5x^7-5x^9+2x^6=-5x^7+2x^6\). The terms \(5x^9\) and \(-5x^9\) cancel. The highest power of \(x\) in the remaining polynomial is 7, so its degree is 7. Option 9 is incorrect because the \(x^9\) terms become zero after simplification. Exam tip: Expand brackets and combine like terms before finding the degree.

Terms with zero coefficients disappear and (-41) remains. It is a non-zero constant polynomial with degree (0).

The degree is the greatest exponent with a nonzero coefficient, while the coefficient of a missing power is zero. We must check both requested conditions: the polynomial must contain a nonzero \(x^{10}\) term, and it must contain no \(x^9\) term. Only option A satisfies both. In option B the \(x^9\) coefficient is 1, and the other options have the wrong degree.

For option A, \(6x^{10}-4x^8+1\), the greatest exponent is 10 because the coefficient 6 is nonzero. There is no displayed \(x^9\) term, so its coefficient is understood to be 0. Option B has an \(x^9\) term, option C has degree 9, and option D has degree 11. Therefore option A is the only correct choice.

First simplify the expression: \(x^5(x^2-6)=x^7-6x^5\). Hence, \(p(x)=x^7-6x^5-x^7+6x^5+29=29\). This is a non-zero constant polynomial, so its degree is \(0\). Option 7 is the highest power visible in the unsimplified expression, but those terms cancel. Exam tip: always simplify a polynomial completely before finding its degree.