Linear polynomials

In \(5y-2\), the highest power of the variable \(y\) is 1, and the coefficient of \(y\) is non-zero. Hence, it is a linear polynomial in \(y\). \(9\) is a constant polynomial of degree 0, while \(y^2+3\) and \(y^3-y\) have degrees 2 and 3 respectively. Exam tip: A linear polynomial must have its highest variable power exactly 1.

In the polynomial 13 - 2z, the value of z can change, so z is the variable. Here, 13 is the constant term and -2 is the coefficient of z; neither is a variable. Exam tip: The letter whose value can vary is called the variable.

Because \(0x^2=0\), this term does not contribute to the degree of the polynomial. The expression becomes \(7-3x\), whose highest power of \(x\) is 1; therefore, it is a linear polynomial. Merely writing an \(x^2\) term does not make a polynomial quadratic—the coefficient must be non-zero. Exam tip: remove terms with zero coefficients before determining the degree.

A linear polynomial has the form ax + b, where b is the constant term. In 2x - 3, the term without x is -3, so its constant term is -3. In 2x + 3, the constant term is +3, not -3. Exam tip: To identify the constant term, look for the term that has no variable.

A zero of a polynomial is a value that makes the polynomial equal to 0. Substituting \(x=3\) in \(x-3\) gives \(3-3=0\), so its zero is 3. In contrast, \(x+3\) has zero \(-3\), not 3. Exam tip: Substitute the given value in the polynomial and check whether the result is 0.

To find the zero of \(x+4\), set \(x+4=0\). This gives \(x=-4\), so \(x+4\) is the correct option. The zero of \(x-4\) is \(4\), not \(-4\). Exam tip: Set a polynomial equal to 0 and solve for \(x\) to find its zero.

\(2x\) and \(3x\) are like terms, so their coefficients are added: \(2x+3x=(2+3)x=5x\). The constant term \(-1\) remains unchanged; hence the simplified polynomial is \(5x-1\). \(6x-1\) would result only if the coefficients added to 6. Exam tip: combine only terms with the same variable and the same exponent.

Here, 9x and -4x are like terms, so their coefficients are combined: 9x-4x=(9-4)x=5x. The constant term 6 remains unchanged; hence the simplified expression is 5x+6. In 5x-6, the sign of the constant term has been incorrectly changed. Exam tip: Combine only terms with the same variable and exponent.

In \(6x\), the highest power of \(x\) is \(1\), so it is a linear polynomial. It has no constant term and can be written as \(6x+0\). Although \(6x+1\) is also linear, its constant term is \(1\). Exam tip: a linear polynomial has the form \(ax+b\), where \(a\ne0\), and \(b\) is the constant term.

Substitute 2 for x in the polynomial: \(p(2)=3\times2+2=6+2=8\). Therefore, the correct answer is 8. The value 6 represents only \(3\times2\); the constant term 2 must also be added. Exam tip: To find the value of a polynomial, replace every x with the given number and then simplify carefully.

Substitute 3 for x: p(3)=4(3)-1=12-1=11. Therefore, 11 is the correct option. The closest distractor, 12, results from calculating 4×3 but forgetting to subtract 1. Exam tip: To find the value of a polynomial, substitute the given number for the variable and follow the correct order of operations.

A linear polynomial has degree 1 and is of the form ax+b, where a ≠ 0. In -3x+4, the highest power of x is 1 and its leading coefficient is -3, which is negative. x^2-3 is quadratic, while 7 is a constant polynomial. Exam tip: To find the leading coefficient, look at the coefficient of the term with the highest power.

For a linear polynomial \(p(x)=ax+b\), substituting \(x=0\) gives \(p(0)=b\), the constant term. In option A, \(p(x)=5x+2\), so \(p(0)=5(0)+2=2\). Option C has constant term 5, so it does not satisfy the condition. Exam tip: To find \(p(0)\), put 0 in place of \(x\).

In the linear polynomial ax+b, ax is the term containing x, so it is the x-term. The term b is the constant term because it does not contain x. Exam tip: identify a variable term by writing the complete term along with the coefficient of x.

(0\cdot x+0) is the zero polynomial and its degree is not defined. Therefore it is not a linear polynomial.

Given \(p(x)=2x-5\), substitute 5 for \(x\): \(p(5)=2\times5-5=10-5=5\). Therefore, the correct answer is 5. The value 10 is only \(2\times5\); the subtraction of 5 has not yet been done. Exam tip: To evaluate a polynomial, replace every \(x\) with the given number and then simplify.

A linear expression in x can be written as ax + b. In this form, a is the coefficient of x and b is the constant term, meaning the term that contains no x. We need a = 4 and b = −1 simultaneously. Option A is 4x − 1, so the coefficient of x is 4 and the constant term is −1; it satisfies both conditions. Option B has coefficient 4 but constant term +1. Option C has the required constant term but its coefficient is −4, not 4. Option D has coefficient 1 and constant term −4. Therefore option A is the only expression that meets both requirements. The signs and the term without x must both be checked, not just the coefficient.

To find the zero, set the polynomial equal to 0: \(6x+12=0\). Thus, \(6x=-12\), so \(x=-2\). On checking, \(6(-2)+12=0\); therefore, \(-2\) is the correct zero. \(-12\) is the constant term, not the zero. Exam tip: substitute the obtained value into the polynomial to verify your answer.

In the polynomial \(6x+13\), the highest power of \(x\) is \(1\), so its degree is \(1\). A polynomial of degree \(1\) is called a linear polynomial. A quadratic polynomial would have highest power \(2\). Exam tip: To identify the type of a polynomial, look at the highest power of the variable.

The governing concept is the degree of a polynomial. The degree is the greatest exponent of the variable having a nonzero coefficient. A linear polynomial has the general form ax + b, where a is nonzero; therefore the greatest exponent of x is 1. Hence option C is correct. A polynomial of degree 0 is a nonzero constant, such as 7, while degree 2 describes a quadratic polynomial and degree 3 describes a cubic polynomial. The word “linear” specifically indicates degree one, not merely an expression containing a variable. Thus the highest power in a linear polynomial is exactly 1.