Degree of a Polynomial

Inside the polynomial, the term with the greatest exponent is x^4. When the entire expression is raised to the power 3, its highest-degree term is (x^4)^3=x^{12}. Hence, the degree of the polynomial is 12. Option 24 is incorrect because the exponents are multiplied: 4×3=12. Exam tip: when a polynomial is raised to a power, multiply its degree by that power, provided the leading term does not cancel.

The term -5/x can be written as \(-5x^{-1}\). Since x has a negative exponent, the expression is not a polynomial, so it cannot have degree 3. Exam tip: check exponents before finding degree.

On substituting \(b=3\), we get \((b-3)=0\). Hence, the term \((b-3)x^{12}\) becomes zero and the polynomial reduces to \(2x^5-9\). The highest power of \(x\) in the remaining polynomial is 5, so its degree is 5. Option 12 is not correct because the \(x^{12}\) term has vanished. Exam tip: substitute the parameter first, remove zero-coefficient terms, and then identify the highest remaining exponent.

Combining like terms gives \(6x^7-6x^7=0\), so the expression becomes \(-2x^4+x^3\). Its highest exponent is 4. In exams, always simplify a polynomial before finding its degree.

First expand the expression: \(3x^4(x^4-2)=3x^8-6x^4\). Thus, the polynomial becomes \(3x^8-6x^4-3x^8+11x^6=11x^6-6x^4\). The terms \(3x^8\) and \(-3x^8\) cancel, so the highest remaining exponent is 6. Therefore, the degree is 6; 8 is only visible before simplification. Exam tip: always simplify a polynomial completely before finding its degree.

Terms with zero coefficients disappear and (43) remains. It is a non-zero constant polynomial with degree (0).

The polynomial must satisfy two requirements: its degree must be 6, and the coefficient of \\(x^5\\) must be zero. Option D is \\(2x^6-3x^4+1\\). The highest exponent is 6 and its coefficient, 2, is nonzero, so the degree is 6. There is no \\(x^5\\) term, so its coefficient is 0. Therefore option D is the required choice.

Option A has degree 6 but contains \\(x^5\\) with coefficient 1. Option B has degree 5, while option C has degree 7 because of its \\(x^7\\) term. Thus only D meets both conditions. An absent power is understood to have coefficient zero in polynomial notation.

First simplify the expression: \(x^3(x^5-4)=x^8-4x^3\). Thus, \(p(x)=x^8-4x^3-x^8+4x^3+19=19\). A non-zero constant polynomial has degree \(0\), so the correct answer is \(0\). Choosing \(8\) would result from overlooking the cancellation of \(x^8\) and \(-x^8\). Exam tip: Always simplify a polynomial completely before finding its degree.

\(p(x)+q(x)=(x^6-3x^2)+(-x^6+5x)=-3x^2+5x\). After the \(x^6\) terms cancel, the highest remaining exponent is 2, so the degree is 2. Exam tip: simplify like terms before finding degree.

The direct answer is option B, \(c=-3\). To obtain degree 0, both variable terms must vanish. Their coefficients are \(c+3\) and \(c^2-9\). Substituting \(c=-3\), we get \(c+3=0\) and \(c^2-9=9-9=0\). The expression is then the constant \(25\), which is non-zero, so its degree is 0. Option A, \(c=3\), makes \(c^2-9=0\), but \(c+3=6\), leaving an \(x^6\) term. Option B removes both variable terms. Option C gives coefficients 3 and -9, so the degree remains 6. Option D leaves a non-zero coefficient of \(x^6\). Remember: the degree of a non-zero constant is 0, but a non-zero term containing \(x\) prevents degree 0.

The direct answer is option B, d=6. A polynomial’s degree is the greatest exponent of x whose coefficient is not zero. Here the x^7 coefficient is d−6. To make the degree fall from 7 to 5, set d−6=0, so d=6. Then the x^5 coefficient is d+1=7, which is non-zero, and the constant term is 14. Therefore the polynomial becomes 7x^5+14 and has degree 5. Option A, d=−1, makes the x^5 term disappear, but the x^7 coefficient is −7, so the degree is 7. Option B works because the x^7 term disappears while the x^5 term remains. Option C, d=0, leaves coefficients −6 and 1, so the degree is 7. Option D, d=7, leaves coefficient 1 for x^7, so the degree is 7. Exam cue: to obtain a lower degree, cancel the highest-power term, then check that the next required term has a non-zero coefficient.

The total degree of a polynomial is the greatest total degree among its terms. For the first term, \(x^3y^4z^5\), the total degree is \(3+4+5=12\). The second term, \(2x^5y^2z^3\), has total degree \(5+2+3=10\), while the constant term \(-7\) has degree \(0\). Hence, the greatest total degree is \(12\). Option \(10\) is only the degree of the second term. Exam tip: In a multivariable term, add the exponents of all variables before comparing terms.

To find the degree with respect to z, ignore the powers of x when comparing terms. The coefficient of a power of z may itself contain x; that does not change the exponent of z being examined. Constant terms and terms involving only x have z-degree zero. The answer is therefore the greatest exponent of z that appears with a nonzero coefficient.

The first term contains z^6, while the second contains z^8. The terms 3x and -1 contain no z, so they cannot give a positive z-degree. Between 6 and 8, the larger exponent is 8, and the coefficient of z^8 is -9x^2, which is not the zero polynomial. Hence the degree with respect to z is 8, making option C correct.

The \(x^n\) terms are the highest-degree terms in both polynomials. If their coefficients add to zero, these terms cancel, so the remaining non-zero sum has degree below \(n\). Constant terms do not control degree. Exam tip: compare leading coefficients first.

Expanding gives \((x^4-5x^2+1)(2x^3+x)=2x^7-9x^5-3x^3+x\). After subtracting \(2x^7\), the highest-degree terms cancel, leaving \(-9x^5-3x^3+x\). Its leading term is \(-9x^5\), so the degree is 5. Degree 7 applies only before this cancellation. Exam tip: simplify fully and check whether leading terms cancel before finding the degree.

On opening the brackets, the signs of all terms in the second polynomial change: \(p(x)=x^6-4x^2+5-x^6+x^2=-3x^2+5\). The \(x^6\) terms cancel, and the highest power remaining is 2; hence the degree is 2. Choosing 6 is incorrect because no \(x^6\) term remains after simplification. Exam tip: Always simplify a polynomial completely before finding its degree.

On expanding, \(x^5(x^3+1)-x^3(x^5-4)=x^8+x^5-x^8+4x^3=x^5+4x^3\). The \(x^8\) terms cancel each other. The highest power of \(x\) in the remaining polynomial is 5, so its degree is 5. Option 8 is incorrect because the \(x^8\) term does not remain after simplification. Exam tip: Always simplify a polynomial completely before finding its degree.

When finding the degree of a polynomial, first substitute the given value and remove any term whose coefficient becomes zero. Here \(e=4\), so the coefficient of \(x^{11}\) is \(e-4=0\). That term disappears completely. The coefficient of \(x^8\) is \(e+2=6\), so the polynomial becomes \(6x^8+5x^2-6\). Its highest power with a nonzero coefficient is therefore 8.

The degree is not 11 because the \(x^{11}\) term has vanished. It is not 2 or 0 because nonzero terms of higher degree remain. Thus the degree is 8, which is option B. The important rule is that the degree is determined only after substitution and simplification, not from the original expression before checking its coefficients.

On simplification, p(x)=0 because every term has coefficient 0. This is called the zero polynomial. It has no highest-power term with a non-zero coefficient, so its degree is not defined. Do not assign degree 0 to it; degree 0 is for a non-zero constant polynomial. Exam tip: simplify a polynomial and remove all zero-coefficient terms before finding its degree.

Let \(\deg(p)=m\) and \(\deg(q)=n\). If their leading terms are \(ax^m\) and \(bx^n\), respectively, then the leading term of the product is \(abx^{m+n}\). Since \(a\neq0\) and \(b\neq0\), we have \(ab\neq0\), so \(\deg(pq)=m+n\). Option B gives the maximum degree, which is not generally true for a product. Exam tip: the degree of the zero polynomial is not defined, so this rule applies only to non-zero polynomials.

The direct answer is option B, \(s=-4\). Degree 0 requires both coefficients of positive powers of \(x\) to be zero. The coefficient of \(x^9\) is \(s^2-16\), and the coefficient of \(x^5\) is \(s+4\). For \(s=-4\), \(s+4=0\), while \(s^2-16=16-16=0\). The expression becomes the constant 7, so its degree is 0. Option A, \(s=4\), makes the first coefficient zero but leaves \(s+4=8\), so degree 5 remains. Option B makes both coefficients zero. Option C leaves coefficients -16 and 4, so degree 9 remains. Option D leaves both coefficients non-zero. A useful cue is to use the linear coefficient first: \(s+4=0\) directly gives \(s=-4\), then verify the squared coefficient.

On subtraction, the like terms cancel: \(A(x)-B(x)=(x^7+2x^3+5)-(x^7+2x^3)=5\). Since \(5\) is a non-zero constant polynomial, its degree is \(0\). Degree \(7\) is the degree of the original polynomials, not of their difference. Exam tip: first simplify the resulting polynomial, then identify the highest power with a non-zero coefficient.

A(x)+B(x)=(x^6+9)+(-x^6+4x^2)=4x^2+9. The x^6 and -x^6 terms cancel each other. The highest exponent in the remaining polynomial is 2, so its degree is 2. Option 6 is incorrect because the x^6 term no longer remains. Exam tip: always simplify a sum of polynomials before identifying its degree.

Substituting a=5 and b=9 gives the polynomial 4x^5+3x^9+1. The degrees of its terms are 5, 9, and 0. The greatest degree is 9, so the degree of the polynomial is 9. Option 5 is only the degree of the term 4x^5, not of the whole polynomial. Exam tip: After simplifying a polynomial, identify the term with the highest power of the variable.

First, expand: (x^4(x^3+x)=x^7+x^5). Therefore, (p(x)=x^7+x^5-x^7-x^5+31=31). Since 31 is a non-zero constant polynomial, its degree is 0. Degree 7 would result only if the cancellation of the highest-degree terms were ignored. Exam tip: simplify a polynomial and combine or cancel like terms before finding its degree.