Polynomials in one variable

\(p(x)=9\) 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 degree of the zero polynomial is undefined. In exams, keep it separate from a non-zero constant polynomial.

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.

In \(z^4-2z+1\), only the variable \(z\) occurs, and its exponents 4, 1, and 0 are non-negative integers. Therefore, it is a polynomial in one variable, \(z\). Options B and C contain a negative power or a variable in the denominator, while option D contains two variables, \(z\) and \(w\). Exam tip: the powers of the variable in a polynomial must be non-negative integers.

The leading term is the term containing the variable with the highest exponent. In this polynomial, the exponents are 3, 2, 1, and 0, so \(2x^3\) is the leading term. \(-7x^2\) is the next term because its exponent is only 2. Exam tip: when a polynomial is written in descending powers, its first term is the leading term.

The governing criterion is the definition of a polynomial in one variable: it is a finite sum of non-negative integral powers of the variable multiplied by coefficients from the specified number system. Dividing the entire expression by 2 gives (x^2 + 1)/2 = (1/2)x^2 + 1/2. This is a sum of x^2 and a constant, and both coefficients, 1/2 and 1/2, are real numbers. Hence it is a polynomial in x, with degree 2. Option A is correct. A denominator does not automatically prevent an expression from being a polynomial; the problem would arise if x occurred in a denominator or with a negative or fractional exponent. Only x appears, so it does not have two variables. Its degree is also defined because the polynomial is non-zero.

In option B, \(\frac{3}{x}=3x^{-1}\). The variable has a negative exponent and occurs in the denominator, so the expression is not a polynomial. The other options contain only non-negative integer powers of the variable and are therefore polynomials. Exam tip: In a polynomial, the powers of the variable must be non-negative integers such as 0, 1, 2, 3, ... .

Substituting \(x=-1\), \(p(-1)=(-1)^2+2(-1)+3=1-2+3=2\). Therefore, option B is correct. A common mistake is to treat \(2(-1)\) as \(+2\), but it equals \(-2\). In exams, always use brackets when substituting a negative value.

Combining like terms gives \\(4x^3-2x^3=2x^3\\), so the simplified polynomial is \\(2x^3+x+5\\). The highest power of the variable is 3; therefore, its degree is 3. In an exam, first simplify like terms and then identify the highest exponent of the variable.

A constant polynomial does not contain a variable, so its value remains fixed. Since \(-7\) contains no variable \(x\), it is a constant polynomial. The other options contain \(x\), so they are non-constant polynomials. In an exam, first check whether the variable appears in the expression.

The degree of a polynomial is the greatest exponent of the variable whose coefficient is not zero. In \\(p(x)=ax^2+bx+c\\), the coefficient of \\(x^2\\) is a, and the question states that \\(a\\ne0\\). Therefore the quadratic term is genuinely present, so the degree is 2. A polynomial of degree 2 is called a quadratic polynomial, making option B correct.

The values of b and c may be zero or nonzero, but they cannot remove the nonzero \\(ax^2\\) term. Hence the expression cannot become linear, constant, or cubic. The condition \\(a\\ne0\\) is essential: if a were zero, the degree could fall depending on b and c. Here that condition fixes the highest nonzero power as 2.