Polynomials in one variable

The expression (3x^2-5x+7) has only (x) and non-negative integer powers. In exams, the variable power must not be negative or fractional.

Answer: option A, 3. The degree of a non-zero polynomial in one variable is the greatest exponent of the variable having a non-zero coefficient. In p(x)=4x³−2x+9, the term 4x³ has exponent 3 and coefficient 4, which is non-zero. The term −2x has exponent 1, and the constant 9 has exponent 0. Since 3 is the greatest exponent present with a non-zero coefficient, the degree is 3. Option B is not correct because the x² term is missing; a missing term has coefficient zero and does not determine the degree. Option C is the degree of only the linear term −2x. Option D is the constant’s value, not the degree. A helpful rule is to arrange terms by powers and select the highest non-zero power, not the largest numerical coefficient.

The constant term is the term that does not contain the variable \(x\). In the given polynomial, \(7x^2\) and \(5x\) contain \(x\), whereas \(-11\) does not; therefore, the constant term is \(-11\). In an exam, identify the term independent of the variable.

Answer: option B, −3. To find the coefficient of x², first locate the term containing exactly x². In the polynomial 5x⁴−3x²+x−6, that term is −3x². The numerical factor attached to x², including its sign, is −3. Therefore B is correct. Option A, 5, is the coefficient of x⁴. Option C, 1, is the coefficient of x because x can be written as 1x, but it is not the coefficient of x². Option D, −6, is the constant term and contains no x. The safest method is to match the requested power first and then read its coefficient; do not choose a coefficient merely because it is the first or largest number in the expression. The negative sign belongs to the coefficient.

A quadratic polynomial is a polynomial whose highest power of the variable is 2. In \(x^2-4x+1\), the highest power of \(x\) is 2, so it is quadratic. Option A is linear, option C is cubic, and option D is a constant polynomial. Exam tip: identify the degree by looking at the term with the highest exponent.

The term (x^{-1}) has a negative power of the variable, so it is not a polynomial. In exams, powers should be like (0,1,2,\ldots).

To find \(p(1)\), substitute \(x=1\): \(p(1)=2(1)^2-3(1)+1=2-3+1=0\). Therefore, the correct answer is 0. Choosing 1 usually results from incorrectly combining the constant and negative terms. Exam tip: substitute the given value at every occurrence of \(x\) and evaluate powers before performing addition or subtraction.

The governing concept is that separate polynomial terms are identified by addition or subtraction signs, while each term includes its sign when necessary. In q(x) = x^3 + 2x^2 − x + 4, the four terms are x^3, 2x^2, −x, and 4. Therefore the polynomial contains 4 terms, making option C correct. The minus sign before x does not create an additional term; it belongs to the coefficient of the term −x. Likewise, the constant 4 is itself one term. Counting only the variable terms would give 3, which is why option B is incomplete. The expression has no hidden fifth term, because there are only three separating signs and four resulting parts.

A linear polynomial has degree 1. In \(3x-8\), the highest power of the variable \(x\) is 1, so it is a linear polynomial. The degrees of \(x^2+1\) and \(x^3+x\) are 2 and 3, respectively, while \(6\) is a constant polynomial of degree 0. Exam tip: identify the highest power of the variable to determine the degree.

The term with the highest power is \\(6x^5\\). The numerical coefficient of this term is called the leading coefficient, so the correct answer is \\(6\\). Remember that \\(5\\) is the degree of the polynomial, not its leading coefficient.

A monomial is a polynomial containing exactly one term. The expression \(3x^2\) has only one term, so it is a monomial. The other options contain two or three terms. In an exam, identify a monomial by counting its terms, separated by plus or minus signs.

A binomial is a polynomial with exactly two unlike terms. In 5x - 7, the terms are 5x and -7, so it is a binomial. Option A has three terms, option C has one term, and option D has four terms. Exam tip: count the terms separated by plus or minus signs, ignoring the sign as part of the term.

A trinomial is a polynomial containing three unlike terms. In \(x^2+x+1\), the three terms are \(x^2\), \(x\), and \(1\), so it is a trinomial. Option B is a binomial, option C is a monomial, and option D is a four-term polynomial. Exam tip: count the terms separated by plus or minus signs.

\(p(x)=9\) is a non-zero constant polynomial. It can be written as \(9x^0\), so the highest power of \(x\) is \(0\), making its degree \(0\). Remember that it is the zero polynomial, not a non-zero constant polynomial, whose degree is undefined.

The direct answer is option C: the degree of the zero polynomial is undefined. The degree of a non-zero polynomial is the highest power of the variable having a non-zero coefficient. For example, the degree of \(3x^2+1\) is 2. But the zero polynomial \(p(x)=0\) has no non-zero term and therefore has no highest non-zero power. By the standard school convention, its degree is not defined. Option A, degree 0, is a common mistake: a non-zero constant such as 5 has degree 0, but the zero polynomial is special. Option B, degree 1, has no basis because there is no x-term. Option C is correct. Option D, degree 2, is also unsupported because there is no quadratic term. Do not confuse the zero polynomial with a constant non-zero polynomial. Exam cue: degree 0 belongs to non-zero constants; the zero polynomial has undefined degree.

Since (\sqrt{x}=x^{\frac{1}{2}}), the variable has a fractional power, so it is not a polynomial. In exams, powers must be whole numbers.

Substituting \(x=2\), \(p(2)=2^2-5(2)+6=4-10+6=0\), so option A is correct. Options B and C result from evaluating a term incorrectly or mishandling the negative sign. In exams, use brackets when substituting a value into every term, especially a negative term.

The degree of a polynomial is the greatest exponent of the variable whose coefficient is non-zero. Here, 3x^3 has the greatest such exponent, namely 3; the term 0x^2 does not affect the degree because its coefficient is zero. Therefore, the correct answer is 3. Exam tip: ignore terms with zero coefficients before identifying the highest power.

A polynomial in one variable must involve only one variable, with constant coefficients. Here both \(x\) and \(y\) occur, particularly in the term \(3xy\), so the expression is not a polynomial in one variable. Having degree 2 or a constant term 1 is perfectly acceptable for a polynomial. Exam tip: identify all distinct letters appearing as variables before deciding whether a polynomial is in one variable.

In this polynomial, the value of \(t\) can change, so \(t\) is the variable. The numbers \(8\), \(-3\), and \(4\) are constants, while \(p\) denotes the polynomial or function name, not the variable. Exam tip: the symbol whose value can change and whose powers appear is generally the variable.

The term containing \\(x^3\\) is \\(-2x^3\\), so its coefficient is \\(-2\\). Remember that the coefficient is the numerical factor multiplying the required power of the variable; the constant term and terms with other powers are not relevant.

A cubic polynomial has degree 3. In \(x^3-2x+1\), the highest power of the variable is 3, so it is a cubic polynomial. Option B has degree 2, option C has degree 1, and option D has degree 0. Exam tip: Identify the highest power of the variable with a non-zero coefficient to determine the degree.

In the standard form of a polynomial, terms are arranged in descending powers of the variable. Here, the powers are \(4,2,1,0\), so the correct form is \(3x^4+4x^2-x+2\). In option D, the term \(-x\) is placed before \(4x^2\), so the powers are not in descending order. Exam tip: identify the power of each term first, then arrange the terms from the highest power to the lowest.

Substitute \(x=0\): \(p(0)=5(0)-3=-3\). Therefore, option B is correct. Option D results from missing the negative sign. Exam tip: for a polynomial, \(p(0)\) equals its constant term.

The \(x^4\) term is absent from the given polynomial, so its coefficient is taken as 0. The number \(-4\) is the coefficient of \(x^2\), while 10 is the constant term. Exam tip: the coefficient of any missing power is 0.