Linear polynomials

The governing concept is classification by the degree of a polynomial. The degree of a non-zero polynomial is the greatest exponent of its variable having a non-zero coefficient. A polynomial of degree 0 is called a constant polynomial, one of degree 1 is called a linear polynomial, one of degree 2 is quadratic, and one of degree 3 is cubic. Since the highest power stated in the question is 1, the polynomial has degree 1 and is therefore called a linear polynomial. Hence option B is correct. Option A describes degree 0, option C describes degree 2, and option D describes degree 3. The classification depends on the greatest exponent, not on the number of terms.

A linear polynomial has highest exponent 1 and is of the form \(ax+b\), where \(a\neq 0\). In \(x^2+1\), the highest exponent of \(x\) is 2, so it is a quadratic polynomial, not a linear polynomial. Although \(\frac{x}{4}-6\) has a fractional coefficient, its highest exponent of \(x\) is 1, so it is linear. Exam tip: To find the degree of a polynomial, identify the greatest exponent of its variable.

\(x^2+1\) is a polynomial because the powers of \(x\) are non-negative integers. Its highest power is 2, so its degree is 2 and it is not a linear polynomial. \(4x-9\) is linear because its degree is 1. \(\frac{1}{x}+2\) and \(\sqrt{x}+1\) are not polynomials because they contain powers \(-1\) and \(\frac{1}{2}\), respectively. Exam tip: in a polynomial, the exponent of a variable must be a non-negative integer.

A non-zero linear polynomial has the form \(ax+b\), where \(a\ne0\). From \(ax+b=0\), we get \(x=-b/a\), so there is exactly one zero. Exam tip: first check that the coefficient of \(x\) is non-zero.

In \(9s+6\), the highest exponent of \(s\) is \(1\), so it is a linear polynomial. \(s^2-1\) is also a polynomial, but its degree is \(2\). \(s^{-1}+4\) has a negative exponent, while \(\sqrt{s}+3\) has \(s\) raised to \(\tfrac{1}{2}\); hence neither is a polynomial. Exam tip: exponents of variables in a polynomial must be non-negative integers such as \(0,1,2,\ldots\).

The point where the graph meets the x-axis represents a zero of the polynomial. Putting \(p(x)=0\) gives \(ax+b=0\), so \(x=-\frac{b}{a}\). Since \(a\ne0\), this is one definite real value; hence the line intersects the x-axis at exactly one point. Option B can occur for a quadratic polynomial, not for a linear polynomial. Exam tip: because \(a\ne0\), a linear polynomial always has exactly one zero.

In \(\sqrt{2}x-\frac{3}{5}\), the highest power of the variable \(x\) is 1, and \(\sqrt{2}\) is a real coefficient. Hence, it is a linear polynomial in \(x\). \(x^2-1\) is quadratic, while \(\frac{1}{x}+2\) has power \(-1\) of \(x\) and \(\sqrt{x}+1\) has power \(\frac{1}{2}\); therefore, these are not polynomials. Exam tip: powers of a variable in a polynomial must be non-negative integers such as 0, 1, 2, ... .

In \(C(x)=12x+50\), the highest exponent of the variable \(x\) is 1, and the coefficient of \(x\), 12, is non-zero. Therefore, it is a linear polynomial of degree 1. The number 50 is only the constant term; it does not make the entire expression constant. Exam tip: find a polynomial’s degree by identifying the highest exponent of its variable.

In \(4x-9\), 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: identify the type of a polynomial by checking the highest power of its variable.

The governing definition is that a linear polynomial in one variable has the form ax + b, where a is nonzero and b is a constant. The highest power of x that actually occurs is therefore x^1, so its degree is 1. For example, 4x − 7 is linear because the variable appears to the first power. A constant such as 5 has degree 0, while x^2 + 1 is quadratic and has degree 2. The number of terms does not determine degree; the greatest nonzero exponent does. Thus option C is correct. Option A describes a nonzero constant polynomial, and options B and D describe polynomials of higher degrees rather than a linear polynomial.

In \(7y+2\), the term \(7y\) can be written as \(7y^1\). Therefore, the highest power of \(y\) is \(1\), so the expression is a linear polynomial. The \(2\) is a constant term and does not affect the power of \(y\). Exam tip: Find the highest exponent of the variable in the terms containing that variable.

In \(2x^2+3\), the highest power of \(x\) is 2, so its degree is 2. Therefore, it is a quadratic polynomial, not a linear polynomial. A linear polynomial has degree 1, as in \(5x-1\), \(x+8\), and \(9-4x\). Exam tip: Identify the degree by checking the highest exponent of the variable.

The general form of a linear polynomial is (ax+b) where (a\neq0). If (a=0), the polynomial will not remain linear.

In \(F(x)=18x+50\), the highest power of \(x\) is \(1\), so it is a linear polynomial. The constant term 50 does not prevent it from being a polynomial. Exam tip: check the highest exponent.

In the polynomial (12-5x), the term containing x is -5x. The number multiplying x is -5, so the coefficient of x is -5. The number 5 alone is not the coefficient because the negative sign is part of it. Exam tip: Always include the sign while identifying a coefficient.

A constant term is a term that has no variable. In \(x+4\), \(x\) contains the variable, whereas \(4\) has no variable. Therefore, the constant term is \(4\). Although \(0\) can be a constant, it is not the constant term in this polynomial. Exam tip: identify the term without \(x\), \(y\), or any other variable.

The polynomial \(2x+7\) has an \(x\)-term and the constant term \(7\), but no \(x^2\)-term. It can be written as \(0x^2+2x+7\), so the coefficient of \(x^2\) is \(0\). The number \(2\) is the coefficient of \(x\), not of \(x^2\). Exam tip: If a term of a particular power is absent, its coefficient is \(0\).

In the polynomial 9x+1, the highest power of x is 1, so the leading term is 9x. Its numerical coefficient is 9; therefore, the leading coefficient is 9. The number 1 is the constant term, not the leading coefficient. Exam tip: identify the term with the highest power of the variable, then take its numerical coefficient.

A linear polynomial has the highest power of its variable equal to 1. In \(3x+10\), the highest power of \(x\) is 1, so it is a linear polynomial. \(x^2-2\) is quadratic and \(x^3+1\) is cubic, while \(7\) is a constant polynomial. Exam tip: To identify the degree of a polynomial, look for the greatest exponent of the variable.

Here, \(p(x)=x-13=1x+(-13)\). The highest power of \(x\) is 1, so it is a linear polynomial of the form \(ax+b\), where \(a=1\) and \(b=-13\). The form \(ax^2+b\) represents a quadratic polynomial because it has \(x\) raised to the power 2. Exam tip: identify a polynomial’s degree by looking at the highest power of the variable.